Melted Charging Plug?

My parents use my old Compaq Presario laptop, a 2500 I think, with a 2.4Ghz Prescott. Now, I knew it got hot, but today while watching the Georgia game, I noticed it wasn't charging (they never take it anywhere, it just stays plugged up constantly.) Everything was plugged in, so I tried to pull the plug...it wouldn't come out. When I pulled a little harder, it did come out, but with no plastic barb on the end, which it used to have, and with metal hot enough to burn my finger.
Immediately, I shut it down, and I'm trying to figure out the best course of action here. I don't want anything to catch on fire, but they have all their pictures and music and other stuff like Quicken files that because of their lack of backup discipline, is not backed up ANYWHERE.

tl;dr: So, I need to get their files off, but I don't really know what my options are with a plug hot enough to melt itself and burn me.
Any help would be appreciated guys.

1. it could have been a short in the plug and not affected the computer, - replace charging unit
2. it could be a short in the computer and totally ruined the charging plug. then you need
file recovery; which is easy -- remove the hard drive and copy it. - HDD enclosure

Thanks for the help curtis!