"It's good to be able to get this done and get on with things," Kane told the club's official website.
"I have enjoyed my time with United and it's good to be able to settle down and hopefully kick on from here."
Cumbrians boss Greg Abbott is confident that the right-back can make an impact next season after sealing his move to the club.
Abbott said: "I expect to see Tony really start to improve and push his way forward now that he has joined us.
"We always knew that he was coming, because it was agreed a while ago, but it is good to have him on board as we prepare for what we hope will be an exciting season."