Hotelwise there are some goodmoptions (depending on your budget). Near the old town, next to the Heumarkt there is the Maritim, on the on theother side of the Rhine, with great view of the Cathedral we have the Hyatt. There is a Hilton near the Cathedral. There is the Marriot Renaissance near the Friesen area, 10 minutes away from the cathredral.
The Guerzenich offers dinner in some of the various rooms, has some live bands and DJs on three levels. It is a great building, but I havent been there for new year for over 10 years when I worked there, so cannot remember if there a lots of kids.
Most restaurants offer vegetable options. The brewery houses will offer some entertainment (i.e. Frueh, Malzmuehle, Pfaeffgen). I also spend new year in a Greek restaurant in Deutz, directly next to the Deutzer Bruecke with some great views of the Old town and cathedral.
You probably have planned already so many activities, but as you are in Cologne over New Year there might be something for you and your kid. The theme park Phantasialand (usually voted the best one in Germany) opens between 24. November and 6. January for an event called Winter Dream. Special decorations etc are used, and is absulte beautiful to go there. Might be a nice option if the weatheris dry.