I would caution against giving your neighbour anything because, as I've said upthread, it could be seen as an admission that you are liable for their loss and it could open the gate (where have I heard that before?) to them coming back to you for more. or for the ridiculous amount they expect you to pay
If you are intent on giving them money, make it crystal clear IN WRITING that you are gifting them the sum of £x as a gesture of neighbourly goodwill and that this gesture should not be seen as any admission of liability or responsibility for the loss of their bike - which a) they should have insured and b) should have ensured that it was secured by a device or chain which could not be broken.
Personally, I wouldn't give them a penny but if you've got money to burn I don't see any reason why you should give them any more than £10 -£20 and I regard the upper figure as being more than generous.
Frankly , it sounds as if you're scared of your neighbour and are kowtowing to him because you find him intimidating whereas, as I see it, he's just another cheeky fucker who should be told to do one.
This is London ffs where no-one in their right mind leaves expensive bikes or other gear outside their house and expects to find it untouched in the next hour/day/week etc.
I live on a leafy avenue in a pleasant part of the capital, but if I'm outside cutting my front garden hedge and need to use the lavatory I make sure that the electric trimmer, shears, and any other paraphernalia comes back indoors with me.
I hope your dc is fully recovered soon.