Go for it! However, keep in mind that it is very crowded till the end of September, when the official tour season ends. October is a good time to visit Leh, with cheaper tickets and accomodations. Overall, Leh needs to be experienced alone. I went there last year by myself and loved it.
Its an amazing place, you would enjoy. Not sure if you would be allowed to go to Kargil, Drass due to the non-peace situation in Kashmir. Rules of Mountains, leave early, reach early. Dont travel at night if you are going my your bike/car.