Bike hire advice from someone running a bike business there.
 

I can recommend a chap called Pete Milburn who regularly runs trips around Shimla and he'd know of any possibilities he's on 01438 832118 mention you got his number from Craig Robinson and good luck maybe see you there if you're still about in July.
Craig - thegreendiver -

Hi,
when in India 4 years ago I first wanted to get the bike in Delhi but since the mountain passes to Ladakh were closed because of snow (beginning of June) and I knew I'd be short of time I decided to get the bike in Shimla or further. I checked the possibility of bike rental being in Shimla and there was none. I don't think that has changed. The first place after Delhi where you can get your Enfield is Manali. Still the bikes that I found in Manali were in poor condition, so it is best to get one in Delhi I guess if you have time.
Regards
Peter