In reply to elliot.baker:
A couple of weeks is fine, three is better.
There are good treks around Ladakh, about five days minimum. Or you could go up to Gangotri (not done it but would like to). If you go somewhere like Leh there are lots of trekking agencies and it's easy to organise. Remoter areas, less so.
Don't forget to factor in getting to the roadhead and back. Travel in Himalayan countries doesn't always run to a timetable. Roads get washed away, towns get shut down by strikes, it can be too cloudy to land planes.
A good trekking guide from LP or similar will give you a lot of logistical information and is well worth the £10 - £15.