It looks like you lifted the pads on the motherboard (need to keep heat on there constantly next time). You look like you have a bit of space to work with there.
Get a scalpel/craft knife (maybe a bit of very fine sandpaper if you have some) and scratch away some of the trace where the pads could go out of to reveal the copper below it- you can solder onto this. The top two on the long side look like you can pull back this way which is nice and although I would want to test it the ones on the short sides appear to be mechanical joins (not a great thing to do with solder but hey) so you could focus on the others if you wanted. The bottom two on the long side are a different matter but if they are not through/multi layer pads then they might not even have to be connected. If they are then we have trouble as you are going to have to find somewhere where they appear and solder wires between the switch contacts and those points.
You can also buy stuff called conductive ink (personally I do not rate it that highly but if you can work it then it is worth having) that can make this a bit easier.
I should apologise- stream of conciousness at 1am concerning matters of electronics is not a good thing for me to be doing.
Yes assuming that post is correct only the top two pads are needed for the DS to be powered/work (you could probably even use some wires if you really wanted to). If you wanted to try and wire it into the holes you can see you probably could, personally I like to repair traces as it is tidier but there is no electrical type reason for doing it in this case.
According to that post there are spare pads on the back as well if you can not get that sorted.
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Or when your player dies, you can say in code, if enemy exists, do nothing, but if enemy does not exist, then create enemy at certain spot. (This would be a pain tho for lots of emeies)