Main problem seems solved but I just want to point out that this is not a super secure or modern lock.

Very unlikely anyone will ever actually try picking the lock on your house, BUT if they did, this lock won't do an incredible amount to stop them.

Unlock the door and remove those screws holding the latch in the door.

Once you’ve removed the lock you should be able to disassemble it and push the stuck key out or worst case, replace the lock cylinder.

Dental pick, like the kind that your hygienist uses to give you PTSD should do the trick.

Use a broken key extractor.

https://www.amazon.com.au/s?k=broken+key+extractor

https://www.ebay.com.au/itm/387398979050

I would remove it from the door, apply sewing machine oil and then slam it onto a surface until a part appears that I can grab with pliers.

I have never done this and it might not work but destroy your lock.

Locksmith here. In a pinch, I've ground the back of a jigsaw blade down, leaving just the teeth on the blade (short enough to fit above the key). Jigsaw blades, especially those intended for metal, are hardened and will easily bite into just about any key, giving enough leverage to pull out the broken piece.

I got a key out of a similar situation in the folks RV the other day using a small flat screw driver and a knife to pinch the key and pull it out...

Dimple lock. The pins move sideways instead of up and down. Not familiar with the brand.

You could try and straighten a small fish hook to remove the broken key.

I've had luck melting a hot glue stick and pressing it on the head, letting it cool, and pulling it out, but you would also be best replacing the lock since you can open the door

While an ignorant rober is trying his luck at trying to open your door a smart one is already gone with your stuff and the police already blasted your door open. I would just get a normal key.

Any time I've had a key stuck like this I've used something sharp and just stabbed it down into the top of the key and leveraged it against the lock itself. Keys are way softer than people think.

Superglue the key back together, let it set, and pull it out.

Be careful not to use too much glue and br sure it's set before you pull.

Alternatively, try needle nose pliers or a hook if you can get any purchase on the key remains.

They also sell extractors but the above suggestions will probably work.

If all else fails, call a locksmith.

Some superglue on the broken part of the key and reattach it, or alternatively superglue a few paper clips or other thin metal objects to the key in the door. Using more than one will spread out the force used to pull.

Why wouldn't you just replace the whole lock?

A neodymium magnet could work, maybe even a fake one from Amazon

superglue.

Strong magnet maybe?