There are plenty of "must see" places in Kenya and 10-day period is just short of seeing all of them. However with that said, you still have a good chance to make practical and effective use of your time.
You can achieve this by either minimizing the number of "must see" places or opting to fly to the various places of interest.
Sample itineraries:
Driving: Spend 1 night Mt. Kenya/Sweetwaters, 2 nights Samburu, 1 night Nakuru and 2/3 nights Maasai mara then fly out to Lamu direct for 2/3 nights.
Option two: Flying
Fly Amboseli 1 night, then to Samburu for 2 nights, then to mt. Kenya/Sweetwaters for 1 night followed by a flight to Maasai mara for 3 nights finishing up at Lamu for 3 nights.
Either of the options above are doable with the latter being a little more dearer in terms of cost.
Tsavo is a little bit out of the way if you still want to get the best out of the Northern circuit, not unless you increase the number of days to 12/14 as franandbrian recommends.
A good tour operator will see you through this critical planning stage. a good example being http://www.e-safaris.com
Most of the places that you have your mind set at are good year round, just weigh your budget and give a shot at a few tour operators mentioned here for price comparison.
Bon voyage!