Hi, I moved back from Nairobi a year ago.
Broadband - patchy and really depends on where you live. Officially, in Nairobi there is complete coverage - many people (and serviced appartments) have excellent broadband. We lived in a very woody area with a leadlined roof and even with a massive booster mast never got more than sporadic connection from the bottom of the garden. However there are internet cafes all over - which are cheap and you can get plugin dongle (which isn't cheap if you use it extensively, but affordable).
There are summer camps, but ours did things via their school. You could google, - there is a good tennis club in I think Lavington, various sports camps in the UN (Gigiri in the north), art club in Lavington. Worth enquiring via the prep schools whether yours could join the sports clubs run there as income generator. I'd guess they'd say yes. An alternative, if you have any contacts there, would be to explore paying the nanny of a family who are on leave to do your childcare. - The summer clubs don't tend to offer wraparound care so even for 11 year olds tend to be morning sessions or afternoon, so you do need an adult to drop off/pick up. Or worth seeing if you could do a driver share.
Also worth considering that the majority of the expat families leave for extended periods over the summer.
Short term accommodation - Yaya apartments have good broadband/pool etc but are in the south of the city; there is lots in the north - again worth looking on the websites of the UN agencies which are in the north of the city or google around Westlands/Gigiri/Spring Valley - though these are more expensive areas. Lots of the hotels run serviced apartments (Warwick centre in North; Fairview in the centre). Alternatively a contact could get you numbers from noticeboards in shopping centres/shops. Or worth putting an advert in the bigger schools (e.g. ISK/Braeburn) to see if a teacher or parent on leave might rent to you.
Daytrips - really worth getting out of Nairobi too for as many weekends as possible. Rift Valley. Malu farm is popular. Nakuru national park, Menengai crater. Early morning game drives. Amboseli for the weekend. Naivasha. The Mara.
PM me if you'd like to know more. Good luck