Hi Laura,
Sounds to me, like you'd want to split your time between the cultural triangle and the hill country. How much you'll be able to cover in that short time would depend on your mode of transport. Also on what time of day you arrive and leave. If it's early in the day you can count that as day 1.
If it's late in the day, then it will probably be a stop over near the airport and start travelling on day 2,
With your limited time, I'd advise going direct from the airport to Sigirya area. Climb Sigirya Rock. Thats what was once a volcanic plug, now just stands as a sheer sided rock with the remains of an ancient palace/ fortress on the top. Very spectacular.
Visit the ruins of Polonorua ancient city, then south to Dambulla cave temples and south again into Kandy.
You should be able to cover those in two days.
One day in Kandy for the Botanical gardens and the temple of the Tooth. Plus the local markets, ( 4 that I know of ) .
That leaves you three days.
Get the train south through the hill country to either Haputale or Ella.
probably Haputale.... Walk up to Liptons Seat on one day and visit the nearby Diyulama Falls on the next day.
Then, guess what,, it's time to get back to the airport. You can do that from Haputale by bus or get back on the train , which will take you back to Peradeniya ( Kandy ) and on down to Colombo. A long ride though, about 8 hours, roughly.
Thats not seeing very much of what Sri Lanka has to offer but it's about all you could do in the time you have available.
Buses and trains are plentiful and cheap but if your budget would cope with hiring a driver and car, roughly about 6000 Lankan rupees a day, you'd find travelling around a lot quicker and probably allow you to include a couple of extra sites.
Rod.