No power at kitchen GFCI. Power at breaker and not tripped?
GFCI outlets can fail. I had a similar issue a few years ago. Replacing the outlet solved the problem. I can't promise that is your problem, but it is worth checking. If you pull the outlet out of the box, and test the voltage feeding the outlet, you should know it that it the problem.
Apologies in advance if you already tried this, but GFCI outlets need to have the power connected before they can be reset. So after connecting the outlet, one needs to push the reset button and possibly hold it for half a second until the thing clicks. Usually, a little green LED will go on. Also again not to insult your intelligence, but it's worth checking that the hot and neutral going to the outlet are on the supply terminals, not the load terminals. Finally, what does your meter say about the line going into the outlet. With the breaker at the panel turned on, try putting your meter on the screws at the back of the outlet, and measuring whether you have line voltage.
test the voltage at the outlet where the GFCI goes. if it isn't 120v, you need to replace the wire from that outlet back to where ever it goes.
Are there/how many other outlets are protected by that kitchen GFCI?....if you have other outlets connected downstream of the kitchen GFCI, you may have another downstream outlet or device that is shorting to ground or bad neutral/ground path wiring defects which prevents the GFCI reset button from engaging and low voltage readings although minor/small voltage readings may just be phantom voltages. Try using an analog meter rather than a digital one which can give false readings Or try a GFCI tester. Check other downstream outlets for loose wiring connections. Unplug any other devices on downstream outlets and try to reset the GFCI...if it works OK, then plug in/power up devices one at a time until it trips.That may be the offending device but some devices may take a while to trip a GFCI.. How many pairs of wires go to the kitchen GFCI? If only one set that connects to the line side terminals, make sure the hot and neutral are connected to the right terminals...installed in reverse /wrong terminals will prevent the GFCI from working too. Since you are getting @120 V in the wiring to the GFCI when disconnected, it sounds more like wiring is reversed or possibly a downstream device/outlet /wiring issue. http://thecircuitdetective.com/gfis.php
It is time to call a licensed electrician. He will be done in less than an hour. They are worth the money.
Roderick has good ideas. Also, keep and read the instruction paper that comes with each outlet. New outlets may have the wires going to different parts than the old outlet. Also, when you reset the panel breaker, turn it all the way OFF before throwing it ON again.
Coffeemakers DO NOT GET PLUGGED INTO GFI OUTLETS as coffee makers turn on and off and makes a tiny spark and the GFI senses the spark and shuts off. You must set the GFI to get it to work. There should be 1 outlet specifically for the coffee maker that is not GFI in the kitchen. The one behind the fridge is only for the fridge. ONLY. It is non GFI and it is a single breaker just for the fridge. If you checked the breaker box, they should be marked as to what each breaker controls so you should find the one that is out AND FLICK THE BREAKER SWITCH TO THE FULLY OFF POSITION AND THEN FULLY ON. yOU CANNOT TELL WHEN THEY TRIP. Then try the outlet reset. Though it is not for the coffee maker. Every kitchen I have worked in has one outlet specifically for the coffeemaker that is NOT GFI.