Exact same situation as in LOTR. Fight begins in Moria.

Dumbledore/Balrog get feats from their movies and books both, but not from offshoots like video games, comics, fanfiction, etc.

Dumbledore has his wand and the Sword of Gryffindor: Balrog has his sword and whip.

Edit: Thanks for all the comments. I hadn't really considered how Dumbledore would attack the Balrog, and a number of people have pointed out that he'd be likely to lean on his mobility, which is significantly better than Gandalf's, as well as his ability to manipulate his environment. Dumbledore's range is a big factor too; Gandalf needed to be right on the bridge to break it, but Dumbledore could easily do it from a distance.

The Balrog is big, slow, and can't attack from distance. His weaknesses are Dumbledore's strengths. Assuming Dumbledore has an inkling of what he's going up against, I think he can 10/10.