Have you asked GDC for recommendations? They are just off Colfax in a somewhat transitional neighborhood where you may not want to hang out with your kid. From that area, you want to look south. The North Cherry Creek area of Denver is very nice, though pricey, and close to where you need to go. There also may be some more moderately priced hotels along Colorado Blvd closer to Cherry Creek or down University closer to DU. Sorry that I don't have any more specific recommendations. I moved to the suburbs 15 years ago and haven't had to look for hotels in that area in a long time.
As for activities, how old is your kid and what kind of interests does your child have? The Denver Nature and Science Museum in City Park is not far from GDC. If your kid likes the zoo, it's right next door to the Museum.
