I know what is most likely wrong. The pins may be damaged, but it is more likely that the charging circuit board is fried. Luckily you can replace this part.
iFixit does not sell the part you need, but you can find the part on eBay. I've ordered from the seller linked below before and their quality is consistent.
iPad 1st Gen Charging Dock on eBay
The board runs from the bottom of the iPad all the way to the top, then it connects to the underside of the logic board. The part actually takes up a lot of space inside the iPad. Here's the guide on iFixit
If you have any trouble you can certainly ask me any questions about the repair. I've replaced this part a couple of times before and it fixed this issue each time.
Also look at this thread for a trick that can sometimes fix the issue