Battery not charging

Sending you a diagram.

It shows fuse numbers and their locations to check.

I also circled a much overlooked item that leaves folks scratching their heads. "Mr. Fusible link"

Many times a fusible link will show bubbled up insulation, be burnt slap into, or since the wire inside maybe burnt in half and not really show externally, if you pull on the (empty) insulation it will stretch out like a rubber band.

You might also perform a continuity test through it.

If indeed you need to replace it, replace it with the same size/rated fusible link. Cheating and just tying the wires together, might leave you watching your vehicle burn to a crisp on the side of the road!

This link might aid you, it might just be a review.

https://www.2carpros.com/articles/how-to-check-a-car-alternator

Let us know what you find.

The Medic

Removing the broken key should not be difficult. It should be able to be removed with a broken key extractor. They look like a straightened fishhook with a blunt tip but the sharp barb is still there. It slides into the slot and grabs the broken part of the key and removes it.

However, it should not be the cause of it not charging. Lights flickering and not charging both sound like you have a bad connection or a bad battery in general. First you need to get a simple voltmeter. The cheap/free units like harbor freight or that you can get online will work just fine.
https://www.2carpros.com/articles/how-to-use-a-voltmeter

Next measure the battery voltage at the terminals themselves. As yours is not charging and it has a draw you may end up needing a new battery first. However for testing, if it will start, use the voltmeter with the engine running, test the voltage at the battery. Next with one lead of the meter grounded go to the alternator and test the output on the battery output post (large red wire). If you see over 13-14 volts at that post but not at the battery you have a break or corrosion in that wire and the voltage is not getting through. That could be a burnt fusible link or a simple broken wire.
Also check for voltage drop on the ground side. Set the meter for the next step over 12-15 volts. Then measure from the engine block to battery ground. and from engine block to the body. Both should read very close to, if not, zero volts. If they do not you also have issues on the ground side which are common with older vehicles as the connections corrode.

If you test at the alternator and see no or very low voltage at the battery terminal and it doesn't change if you rev the engine then it is possible the regulator inside the powertrain module has failed, that requires replacing the PCM and having the new one flashed to match your vehicle if needed.