To narrow down your search for resorts which have kids clubs, I have done a little research and found these ones to be rated as some of the best on the island. You will need to check out each of their websites to find details about ages, open times, and if they are free or have an entry fee.

Courtyard by Marriot
Jl Camplung Tanduk , Dyana Pura , Seminyak.
Kids Club for ages 4 – 12 yrs.

The Grand Nikko
Nusa Dua
Jungle Camp ages 6 – 12 yrs.

St Regis Bali Resort
Kawasan Pariwisata Btdc, Nusa Dua Lot S6, Nusa Dua 80363, Nusa Dua.
Children’s Learning Centre from Age 4 – 12.

Novotel Bali Nusa Dua
Jl. Pantai Mengiat, BTDC Complex | PO Box 116, Nusa Dua
4 – 12 years old.

Padma Resort
Jl. Padma No. 1, Legian
Kids Club from 2 years old and Childrens Pool.

Sofitel Beach Resort
Lot N5 Nusa Dua Tourism
Children’s Club for over 12s Children’s Club for 4 to 6 years Children’s Club for 0 to 3 years Children’s Club for 3 to 6 years Childrens Club for 6 to 12 years Babysitting on request.

Holiday In Baruna
Jl Wana Segara, Tuban
Kids Club and Teen Room, & kids pool.

Bali Dynasty Resort
Jl Kartika South, Kuta
Kids Club for 4 – 12 years, waterslides, trapeze school, and activites.

Ayana Resort 
Jimbaran Bay
4 – 10 + a Kids Pool.

Bali Mandira
Jl Padma, Legian
3 – 12 years old.

Grand Mirage
Tanjung Benoa, Nusa Dua
3 – 12 years, play ground.

Club Med
Nusa Dua
3 – 17 years.