No direct experience of KG itself (though we used to live v close, we moved before school... we did research local schools, and all the kids we know are in y1 atm). It is large compared to many schools locally, has a diverse population and a large catchment. It also has a large population of English as second language. The parents also park like I haven't seen anywhere else (so the main road can be insane).
That said, I have heard a lot about the head being great, and they really have a decent reputation for SEND and helping bring children on - the demographic is just much more mixed than the others! I know a couple of parents who are happy with the school and the progress their kids have made/are making.
Most local kids we knew chose either St Barts and Elliot Bank over Kelvin Grove. Both of these schools are easily walkable if KG is your nearest school (one is up Kirkdale, the other is down Kirkdale and left). St Barts is generally oversubscribed in recent years with a shrinking catchment, but the way London primaries go with the flight of families during primary years, you can never really tell whether there will be a space.
Holy Trinity is also walkable if a Catholic school is their thing. However, the one person I knew whose child went there wasn't very enthusiastic about it at all, and ended up transferring to KG as they had spaces. Adamsrill may also be another option- it's another one I have heard mixed feedback on.
Can't tell you much about Hasletine, St P and Neri, St Michaels etc in Sydenham, and it's unlikely you would get into Horniman in FH from there as catchment is tiny and they tend to stay. Possibly Fairlawn in FH could have spaces- only thing I recall is that they don't have uniform and I have friends who chose St B over Fairlawn! Depending on where Dulwich Woods, possibly, but not sure what the situation there would be, and it could be a long trek.
(Added extra as it sounded so negative!)