Hi,
It seems that you have put yourself in a rather nasty situation.
First off you need to evaluate if the connectors on the board have also been damaged, if one or more have then you have already reached a point where it would be better to connect the 5s to iTunes without the screen and do a backup.
Once you have a backup you can look at buying a replacement 5s, as they are relatively cheap now and compared to having to get a micro soldering shop to replace board connector/s and/or potential board failure with the screw holes, it’ll be the cheapest way out.
You won’t know if the new screen is damaged or not until you can get another device to try it on, but at least you already have a new battery for a replacement 5s.
Probably not what you want to hear but without access to another 5/5s you won’t know.
Hope this helps you to decide what to do.