I recently poked a hole into my freezer while chipping away at the iceberg inside.?
Can my unit be saved?

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Sounds like a visit from the repair guy is in store for you.
You broke it - replace it, and don't defrost the next one in that way.
L
The freon is out, you need a new cooling system.
IME the evaporator coils (the bit you punctured) can't be repaired, they're usually very thin steel, and have to withstand quite a high pressure.
Steve
It's a closed loop, any pressure released WAS the freon (or other more modern refrigerant).
Repairing the hole is possible but not economical given the cost of the unit + having a service person recharge the gas.
You could convert the compressor to an air compressor or vacuum pump, otherwise it's a write off.
***I know the above, because I've done it to a bar fridge myself. I thought I was being very clever getting the inch of ice out with a flat screwdriver and a hammer...until I heard the hiss of death, and it wasn't minecraft.