Hi @nurarockway ,
Seems as though there are two problems.
Firstly, as you say the centre pin of the DC-In jack has broken off, then the jack will have to be replaced as it cannot be repaired. The supplier in the link is an example only to show the part and what it may cost. There may be other suppliers online that suit you better. Just search for MSI GE72 6QD DC-In jack to get results.
Secondly, given that the laptop still works on the AC power it appears that there may be a problem with the battery charging circuit on the motherboard that was caused by what occurred to the DC-In jack.
You would need to have the schematics for the board to find out which components were affected, unless of course it is obvious to see what damage has occurred on the motherboard e.g. burnt out components etc. I couldn’t find a free schematic online, you may have better luck.
In saying this if it were obvious what may be hard then is to know what type of components they are and their values etc. Even if you know the type of component due to their board designation e.g. Q = transistor, R = resistor, L = inductor, C = capacitor, D = diode etc then if it is burnt out then you might not be able to make out its value. That’s why the schematics are needed.
If you can’t read schematics or don’t have a DMM (digital multimeter) for testing purposes, contact a reputable, professional laptop repair service and ask for a quote.