Okay...the 'long' answer: Water comes from many sources. I'll run down the most common: Plumbing leak - if this is a single family home structure that's easy enough to determine whether applicable - I cannot 'see' your place so you'll have to determine that.

You don't indicate whether this is a condo (multi-floor structure), a multi-story structure of any kind, a duplex, etc.

You cannot tell just by 'looking' it takes some sleuthing - it's a kitchen which means it has many water source potentials. Most commonly it's a roof flashing where a sink vent, stove exhaust, or some other utility penetration (like a hot water heater exhaust). But could also be a water pipe, an icemaker pipe, heating/cooling drain pan.

You say there's 'no space' up there - so that tends to rule out HVAC pan/stuff. But the other elements are possibilities - as are adjacent room services (bath, shower, water closet, laundry room).

Just because it stopped doesn't necessarily mean it's a roofing/weather problem - but most likely.

As the water built up above the ceiling it eventually saturated and perforated enough to have water evidence inside the space. Prior it was contained above, and building up a volume of water. When you poked it drained out a majority of the 'reserves' - it still may be leaking from any source...just not enough for it to be leaking through the roof. It could still be soaking into other materials. You've just changed the water path. When you poke a hole you potentially create a 'mountain' on the other side - where the water cannot continue to leak out of the screwdriver poke through spot...and may be saturating/finding another location - gravity always working on the water.

I cannot see your roof...but if it is a roof leak - these are usually pretty small. Buildings expand and contract - different elements at different speeds/distances. Thus the problem is usually at a penetration.

Your roof consists (most likely) of 3 layers...the upper protective/sun/water layer - this could be tile, shingles, rubber, tar rolled - lots of options here. Below that you have the vapor barrier or tar paper...and below that the roof sheathing (plywood most commonly).

Water, mostly wants to go 'down'...as many have said it can leak in one spot and travel along a pipe, conduit or rafter through adhesion/surface tension, and migrate to a spot where it has to drip down...and eventually start to puddle.

You cannot assume it's just the roof from what you've said - you'll have to get up there and find out the reason.

I'm assuming your home is a flat roof with nothing above it (I would hope you'd be smart enough to share/consider other leak sources - like an adjacent tenants toilet or piping...so I'll assume this is 'your house' and a single family dwelling just without an attic. No attic does make things harder.

If you're certain you've located the leak - many posts here have discussed how to locate and how to remedy - so I'll not waste their efforts repeating.

As for insurance - they're always looking for a reason to say "NO" or an avenue to decrease their obligation. It's through documentation you prove this later - and be certain your claim adjuster and EVERYONE knows you're collecting evidence - and retaining it. Your policy either covers this event, or it does not - I cannot answer that. If it does cover it they'll be looking for a way to save money or pay nothing. Showing good faith IMMEDIATE effort to remedy goes a lot farther than 'doing the wrong thing' - But wrong efforts which further cause damage could be held against you. The sooner you can get professionals out there - PAID AND UNDER CONTRACT the better. Just having them show up and review does show good faith but that company isn't responsible until you engage them through payment/writing.

Consider the potential for further harm to the structure and take steps to remedy that. That is what 'they' will use against you. Client learned of leak and DID NOTHING ABOUT IT for X # of days. That's wording that will look bad for you. You saying: I discovered the leak and 20 minutes later we sourced the location and put up pots/pans or hit it with silicone and called professionals would look great for your side. Chances are though, if you're covered the insurance will just pay without incident...these stories of them not paying usually has some mitigating factor or the 'real story' that wasn't told. Like a roof that knowingly leaked for months.

Anyway water leaks aren't rare. It could be anything from blown water in/under tiling...to ripped up shingles due to wind/animal...to a very old roof needing replacement...or something as small as a TINY crack that occurred at a roof vent...and is rolling down the roof vent to a collection spot, collected and eventually showed up inside at the kitchen.

I don't know your age, ability, tools on hand, or mechanical capabilities so I'll not 'instruct' you to do anything. Mostly if you can locate the leak caulking is enough to solve the problem for the short term until pro's can come fully remedy. If it's a leak that you cannot spot tarping the whole roof may be needed. Again, too many complexities there for a 'simple discussion' on that. Tarps also leak...and the potential for blown in water still exists. As does rooves are large areas where tarps are big and heavy. And without knowing the size, and weight abilities of your roof I'm not going to advise you to go up there. If you do choose to do that, be safe...use proper safety gear. And remember, that tarps work like shingles, remember to start at the bottom and leave plenty of overlap if you need multiple tarps.

I don't know the 'strength' of your roof...or how long this saturation has happened. Roof sheathing is very strong even when wet...but long term rotting can lead to structural failures and walking on a roof can considerably further damage a poor situation. Make sure as much of your inspection happens from below prior to you considering going on the roof. For all I know you're 72 years old and don't even belong in the category of even pulling out a ladder.

So don't assume it's 'rain' and don't assume that just because it 'stopped' it's because it's not raining. The hole - as many have said - or the problem will NOT FIX itself. Once the liquid barrier is broken it'll keep leaking until the water source is removed.

For all I know you've got a poorly insulated line up there and you're getting condensation drips - thats where warm/moist air hits a cool surface like water drops on the outside of a cool drink. Ultimately there's water where you don't want it coming out and it won't 'fix' itself without you causing that to happen. If it's a leaking water pipe just create some runoff path - something as easy as a string tied for the water to follow is enough. If it's a pressurized line that you can shut off - do so and turn it on as needed to minimize downtime leaking. If it's a line that can be unused (such as an ice maker feed) - just suffer without it for a while.

As you said there's no space - I'll assume you know what you're talking about. But common leaks are from pans, drain lines, or overflow lines from HVAC units. There's also a secondary pan below the unit that can also leak. Where I live/work I hear people all the time say 'we do not have an attic' - but there is a crawlspace. To them attic means something they can climb up into and store stuff via a ladder or stairs. If it's just 'roof above' as you say it makes tracking down the leak source easier - or harder depending on viewing angles available. Ultimately you have to find the leak and plug it and determine how best to proceed depending on what is leaking, how, and why. Good luck.