Costa Adeje Palace in Southern Tenerife has an indoor pool and a (very well) heated outdoor pool. It is midway between Playa del Duque, one of the nicest beach, and la Caleta, small village with plenty of restaurants.
The weather can be a hit and miss, so you want to be prepared with an hotel with bad weather options.
It you are after more guaranteed sun, I would look at deals in the Red sea area, either Egypt or Israel
When DS was younger we enjoyed staying at the Regency County Club in Tenerife. It has 1-3 bedroom apartments but also access to a nice restaurant and hotel facilities as well as mini golf and a small kids club that your DD would be a perfect age for.
I'm with you on hating formal dress. When you spend your life at work wearing smart clothes you don't want to be told what to wear at a hotel. However most hotels require long trousers at least for dinner through as though it keeps out the riff raff . With DH and adult DSs they take a par of long trousers and wear them just for the meal. Though to be fair, in February it won't be shorts weather in the evening anyway.
It's proving really hard to find something suitable. Nearly booked Roca Nivaria Gran but then noticed it's formal dress code for the restaurant and I really don't want poor DH to have to put a shirt on for dinner.
In February I would stick to southern Tenerife. Fuerteventura is lovely but very windy even in summer and can have unreliable weather in February. I was in Lanzarote in March this year and it was quite cool but we had an apartment that was fabulous. Lots of apartment complexes offer 2/3 bedroom apartments with B&B or HB. I wouldn't go HB myself as there are infinite numbers of great restaurants.
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