Is there any way to save my mini fridge after I punctured the line?

The short answer is that your fridge is officially dead. When you chipped at that ice and heard the hissing, you punctured the evaporator plate. That hissing was the pressurized refrigerant escaping into your room. Once that gas is gone, the compressor can run all day long but there is nothing left to acutally absorb the heat and cool the interior. Electrical tape or duct tape won't do a thing beacuse the system operates under significant pressure, and even if it did hold, you've allready lost the chemical charge needed for the cooling cycle to function. Repairing a mini fridge in this condition is almost never worth it. You woud need a licensed technician to braze the hole shut, vacuum out any moisture or air that entered the lines, and then recharge it with specific refrigerant. The labor and equipment costs alone will be double or triple the price of a brand new unit. Plus, if it was a newer model using R600a (isobutane), that gas is actually flammable, so you shold keep your windows open and get that unit out of your living space as soon as posible.

And for future reference, never use metal tools or anything sharp near those freezer plates. Those aluminum lines are incredibly thin and soft. The only safe way to defrost is to unplug the unit, put some towels down, and let it melt naturally. If you are in a rush, a bowl of hot water or a hair dryer on a low setting is okay, but even then you have to be careful not to melt the plastic liner. But as for this specific fridge, it is time to start looking for a replacement on the used market or at a local scratch-and-dent appliance store.

Honestly, just toss it and grab a Midea 3.1 Cu. Ft. Compact Refrigerator. I switched to thier double-door model after a similar disaster and it's way better because the freezer is actually seprate. You won't get that massive ice shelf buildup that forces you to chip away at it.

They're super cheap at most retailers and way more reliable than trying to patch a dead compressor system. Plus, the Midea runs quiet enough that it won't drive you crazy if it's in your bedroom. Definitely worth the upgrade over another single-door unit.

I’m not totally convinced it’s a goner just yet. Have you ruled out the possibility that the electrical tape is just covering up the temperture sensor bulb? If you patched right over that curved part of the liner, you might have just insulated the thermostat from the actual air temp. I'd also check to see if the compressor is even vibrating; have you ruled out accidentally knocking a wire loose while you were panicking with the wrench?