iPhone 5S - Apple Logo, & Black Screen Shows Up Stuck

The iPhone 5S will not boot up without a battery; it just shows the Apple logo and then restarts again. You need to have a working battery, even if it's not in perfect condition.

Looking at your question...

Also I tried unplugging the battery port from the motherboard then charging it and booting it, but it still doesn't work.

What do you mean by that? Did you take out the battery and charge it independently of the device? That sentence is not clear enough for us to understand.

@oldturkey03 is correct. You need to isolate some variables and work with known-goods, especially with water damaged devices. The fact that iTunes detects your phone tells me the charge port assembly is most likely okay. The battery is an obvious element to replace but because you have very old water damage to the device, I would suspect a logic board issue, most likely the charge circuit.

Did you remove the shields when you cleaned your logic board? If not, that's where 80% of the electronics resides and most likely where the problem is. At that point, you need to de-solder the shields.

So I would try the battery first. Then if that doesn't solve your problem, you should consider a shop that can do micro-soldering repairs to do the diagnostic for you.