I post specific logic board questions here sometimes and surprisingly, I usually get a specific response. But a really good forum that has taught me a lot is the gsm forum. Really great stuff over there.
Its soldered on to the motherboard, I would imagine most shops would charge between $70-$120. However, it is not a repair that every repair shop will know how to do.
So you removed the original glass and then put it back afterwards? Are there any small tears in the digitizer cable? Did you use a metal tool to lift the digitizer connector "flaps"? I would imagine that the screen could have been damaged in the removal process, it is very easy to do so.
It probably needs a new power button. Power buttons for the s3 were failing all the time and now its the s4 turn because they used the same button. I have to fix about one or two s4 power buttons a week now. Boot looping, or vibrating right when you put the battery in usually indicates thr power button. Unfortunately it is soldered to the motherboard but it isn't a difficult job.
Does it charge very slowly when it is on? If so it might be the charging ic on the motherboard, in which case you might have to send it out to someone who can micro solder. Had you tried more than one charge port? It could also be a faulty charge port or battery. Most aftermarket batteries are very terrible as well so it can also be that if you dont know or trust the source.