Plug in by ethernet.. check your computer gets an IP from the TC .. if not you will have to hurt your fingers even more.. but I suspect you have broken it.
You do not need the wireless password but you do need the admin password.
No admin password no reset via software.. you must do it via the switch.
The reset button is a plastic lever.. not direct connection to switch on the board.
I think it is important enough to give you picture.

The switch on the board is between the metal can of the ethernet port and the kensington lock port.. Notice on the lever arm a dimple.. that is where your hole to the outside world comes in.. so about half way between the plastic hinge at the top and the switch at the bottom..
Please notice what this means.. you need twice the force you would need to press the actual switch.. because the lever arm has half the movement. But people use massive force.. and I have opened the airports and TC and found people have bent the lever arm right over the top of the switch.. and jammed it forever in place.. if the plastic button in the hole is missing.. and you are seeing nothing.. then someone has done that..
Assuming the lever is still there..
Press the lever with an opened paper clip.. it takes a fair bit of force.. and the click of the switch is easy to hear in a quiet room.. apply no more force once the switch has clicked.. it is very positive. If you hear nothing or the lever doesn't move.. it is broken, bent or not there.. you will need to disassemble the TC and get into it with a real screwdriver.. and that also means mains voltage so be very very careful.
Here is the full reset method. The flashing amber is double speed.. ie flash every 0.5sec instead of every second. It is double the speed of normal.
Factory reset universal
Power off the TC.. ie pull the power cord or power off at the wall.. wait 10sec.. hold in the reset button.. be gentle.. power on again still holding in reset.. and keep holding it in for another 10sec. You may need some help as it is hard to both hold in reset and apply power. It will show success by rapidly blinking the front led. Release the reset.. and wait a couple of min for the TC to reset and come back with factory settings. If the front LED doesn’t blink rapidly you missed it and simply try again. The reset is fairly fragile in these.. press it so you feel it just click and no more.. I have seen people bend the lever or even break it. I use a toothpick as tool.
N.B. None of your files on the hard disk of the TC are deleted.. this simply clears out the router settings of the TC.
You cannot reset from the airport utility unless you use ethernet.. and know the admin password.