My key is stuck in the ignition of my Impala 2014 and the car won't fully start. (It'll light up the interface and nothing more) ?

Go to a locksmith.

He has a tool to get the broken piece out.

If the car won't run, then find a mobile locksmith, or tow the car to a regular locksmith.

Right now you're stuck.

Buy your way out of the obvious problem.

Usually when this occurs, you have to replace the ignition lock (what the key goes into). Not too difficult and most come with new keys even if it's a VATS key system. Just disconnect the battery if you have to pull the steering wheel (old school ignition locks) so the air bag doesn't go off by accident and break your neck.

You have 2 SEPARATE issues.

1. You can't remove the key.

2. When you turn the key, the starter doesn't seem to engage.

You have already received good answers to the first issue.

For the second, you need someone to inspect the car to diagnose the cause.

Turn the steering wheel hard left or hard right and try to remove the key.  Lube the key slot and let the lube soak for 30 mins and try again.  No luck, time for a professional.

Lubricate the keyhole and wiggle the key. Did you put in the wrong key and now it's jammed? You might have to dismantle the lock cylinder to get the key out. Call a locksmith. Don;t call me a locksmith, I'm a sniper.

Try WD40.   It will loosen the key.  Or a locksmith.

Turn the key to the ON position and work the shift lever back and forth in both directions several times.

Then put it in PARK and see if the key comes out.

If it doesn't come out you my have a solenoid acting up that will need replacing. I know older GM vehicles had a small access plug on the bottom of the steering column that could be popped out and a screwdriver could then be inserted to release the key manually.