I'm assuming that the class is for the 10 year olds as well? I would break them into smaller groups (maybe 4) and rotate stations, assuming you have enough parent volunteers to man each rotation. Bringo sounds fine. I would also do one where you split the rotation group into two lines and have them do a gift-wrapping relay. You have an empty un-assembled box, and wrapping supplies. So the first person in each line has to assemble the box and tape it up. The next person has to wrap it in paper, the next ties a ribbon around it, next puts a bow, etc. The first line to finish wrapping wins.
Kids also love games like cotton ball in spoon races (also done in lines) or something where you have to tape cotton balls to someone's face to create a Santa beard (also done in lines- I love the line races, and the kids also love that). Hope that's a start. These should be able to be done without moving desks around. Things like bingo are great, but too much of those or too long they start to get restlesss. Oh, you could also just have a cookie or ornament decorating rotation- endless possibilities of crafts and snacks there.
For the gift- I know those hats that come down a bit on the sides with ropes hanging off each side are very trendy right now. Even ones with little animal faces. Go to any Target or Kohls or whatever and you'll see a bunch, easy enough to find a blue one. I'll bet the kid has an iPod, or at least a family computer, so I would think iTunes are a safe bet. Even without an iPod, you can listen to the music on a computer. And I'd guess if the family can afford to send her to dance they have at least a computer if not an iPod for her. Otherwise some cute and fuzzy slippers. Or maybe a tote bag that she can keep her dance stuff in? Do you have a Justice store by you? It is tween central, and has everything that age likes. If you don't, look at their website www.shopjustice.com for ideas. Oh- how about blue nail polish?
Good luck!