where can I go
 

where can I take my 11 year old daughter to a great hotel with lots of stuff for her to do and maybe also a golf course for my husband, somewhere hot in Janruary,

Many places. What's your budget? Does that include air? Where from? How many nights?

During peak week? (Christmas thru new years?)

welcome to fodors your majesty. i think you might enjoy barbados. far enough south so hot weather is gauranteed as well as warm water. has 6 golf courses, beautiful beaches and many activities. what kind of things does the princess enjoy? would need to know your hotel budget to suggest someplace.
http://www.barbados.org/childrens_activities.htm
http://www.barbados-beaches-plus.com...-for-kids.html

You have to also define "hot" - Caribbean daily high temperatures in January are typically in the low to mid-80's F. If that's not hot enough for you consider going further south to parts of South America.

Honestly, your question is much too general. There are literally scores of islands, hundreds of "great hotels" with access to golf courses throughout the Caribbean.

You have to give us more information about your likes & dislikes, what type of island experience you are seeking and your budget (we need a dollar number, not a nebulous term like "not too expensive" or "affordable").

You should know that January is peak season in the Caribbean so you are going to encounter some of the highest room rates of the year. If you are planning to travel during Christmas and/or New years then in addition to high rates you may encounter minimum stay requirements.

Atlantis Resort in Nassau (the Bahamas) is very popular with families with children your daughter's age but it is somewhat far north (it's not actually in the Caribbean) and can be cool. You might want to look at Beaches resort in the Turks & Caicos - it's also very popular with families.

Have you considered Disney's island........ fun for everyone.