the connections on the logic board were bent and when i tried to bend them back they broke...

This is a common occurrence in older phones because the batteries do not last as long as the rest of the phone. A simple Battery Replacement should do the trick!

It is likely you cannot easily replace the housing because it would require you to take the entire phone apart and rebuild it in the new housing. A repair shop may be able to help but it is likely going to be expensive.

This is a problem I had on a 6 plus last year. Apparently the 6 plus is prone to "bending" which causes the LCD connectors to break or not connect correctly. I took my phone to apple and they said it would be $300 to fix it or I could get a new phone. You may want to ask a repair shop that can solder.

I would take it to a cellphone repair shop where they can use small tweezers to remove it. If you try prying it out that may cause damage to the housing on the isinde.

You may need to try a different charger there could be a short in the cord. Try and different cord. If this doesn't work it is most likely your battery which can be purchased for around $25 for the part.