I didn't have kids when I visited KK many years ago for the first time. I joined one of those local day tours to Kinabalu National Park and Poring Hot Spring (like this one: click).
From my experience, not much walking/hiking is required. Perhaps an hour or so in the national park. We enjoyed the visit thoroughly. But the problem with small kids, as I foresee, is to have him sit properly during the 2-hour drive to the desination.
So, two years ago when I brought my boy to KK (he was just 1.5 years old), we didn't go uphill to Mount Kinabalu. We went to visit the islands for two days instead.
I don't think children are able to climb up to the summit. However, you may want to ask the guide to take you to a lower and easier track. It is worth to take a trip to Mt. Kinabalu park, which is 2 hours drive from the city.
If you really going to climb to the summit, please follow the tour-guide because there were real cases where some foreigners got lost and were never found.
For more information of Mt. Kinabalu, you may see below link:
http://www.geocities.com/RainForest/6293/