I'd taxi to Santa Monica. I stayed here with DS a couple of times. http://www.loewshotels.com/en/Santa-Monica-Beach-Hotel/index/about You can walk to the pier (has a nice carousel), small aquarium, beach across a small street. Walk (with stroller) to shopping, other restaurants. Nice restaurant in hotel. Definitely kid friendly.
Yeah stay over near the beach....there are some cute beachfront places in Malibu.......maybe google,but you would need a car. If no car maybe marina del ray would be fun with kids as there are all the boats and it is soon close to LAX.santa Monica good too but further from LAX.
How about Manhatten beach which is about 10mins from LA airport, could be a good place to wait for DH? I stayed at The Belamar. I think if was around $200 dollars a night. Has a pool and a shopping mall that is walkable from the hotel. The beach was around 20min walk, but quiet and places to eat nearby. You can get a bus from LAX or a taxi was only around £15 dollars I think.