With so little time you should stick to the Leh area. Your best bet is probably the Markha valley, which is quite close to the Stok Kangri by the way. Be aware that it's the most frequented trek in Ladakh, so you won't be alone on the trail. You could do it in 5-7 days if you drive to Chilling. Alternatively you could do Lamayuru-Alchi (5 days walking), but it implies a 1 day drive to Lamayuru and to be well aclimatized. For more infos check the Trailblazer guidebook "Trekking in Ladakh".
There are plenty of trekking agencies in Leh, but none that I can recommend since I went just with a guide and ponyman I knew from a friend of mine.