Torne Camping rents canoe on the northern part of Åsnen. The map pin is the campsite in Torne. Even if you are not staying at the campsite you can rent according to their rental page.
What they offer
On tornecamping.com there is aluminium canoe with paddles and life jackets. A barrel and a canoe trolley are mentioned as extras.
Good to know
Åsnen is a large lake with many islands. Always wear a life jacket. Current price and season are with the campsite.