OK- You might consider a variation on this-
Fly into Glasgow- stay 2 nights
train or drive to Inverness- 3-5 nights with day trips (lots to choose from- Culloden Battlefield and Clava Carins, Eliean Donnan castle and Isle of Skye, Great Glen, Glen Coe, Ft. William, Ft. Agustus, even Orkney- though it is a long day, whisky tours, cruises on Loch Ness and the firth, etc.) There are lots of reliable tour companies that base in Inverness. I'd suggesst going with one that does small van tours rather than big bus (coach) tours. We have used Puffin Tours, Canny Tours and similar ones.
Then train to Aviemore- 2 days with day trip to Blair Athol castle and maybe 1 or 2 distilleries, there is also a microwbrewery that does tours and gives samples in town, a restored steam train, and the funicular up Mt. Caringorm- lovely views on both of theses, as well as hiking, etc. if interested.
Train to Edinburgh- 1 or 2 days
train to York- 1 night
train to London- 2 or 3 nights (or skip York and London and fly home from Edinburgh- or tani back to Glasgow to fly home)
You will need to tweak your days to fit to the 13 days (does that include flying over and home- if so major cutting needed). Personally, I'd cut out Glasgow and reverse the iternerary.
Feel free to private message me. We've been to the UK more than 40 times in the past 20+ years.