Lake Louise is inside the Banff National Park, one of the best known places in Canada. Full list of activities can be found in the park website and in the website of the Lake Louise and Banff towns.
World-class skiing can be found at Lake Louise Ski Resort accessed by a complimentary shuttle bus that leaves from the hotel. Rentals available at the Chateau, the ski resort and the Lake Louise town.
Cross-country skiing in the area is superb. There are trails accessible directly from the Lake, and a few more from the town (Pipestone trails). Rentals available at the Chateau or the town.
Winter walking can be done with boots in packed trails (might need ice studs), and with snowshoes otherwise. There are hikes that can be accessed from the Lake and, if you have a car, you can go further and explore those closer to the town of Banff, like Johnston Canyon. Trail conditions are updated regularly by Parks Canada on their website.