It is doable but daylight will be in short supply and is likely to be cold, wet and windy (especially in the west on Skye). Have a look at a map of Scotland and you will see that you have two main routes to the Highlands, the eastern route via the A9 or the western route via the A82 and A87 which could take in Glencoe and Fort William.
If you are certain that you want to come to Skye then bear in mind that accommodation providers such as small B&Bs will likely be closed by the time you take your trip, so your options will be limited to the few hotels on the island. You will want to ensure that you can get accommodation so I'd plan that part of your trip first and build the rest around it. Skye is a 3 hour drive from Inverness and it takes about 1.5 hours to drive the length of the island. Many people underestimate the size of the island and the amount of time it takes to drive around here, as the roads are narrow (often single track with passing places) once you get off the main A87.
I'm sure the distilleries over in Speyside will still be doing tours and you could also check out Talisker on Skye. Note that WhiskEy is Irish and Whisky (without the E) is Scottish.
Hope that helps a little.