The 27-year-old has been training with the Cumbrians since the beginning of July after being released by the Scottish Premier League outfit.
"Lewis genuinely wants to be at the club because he has told us it's the happiest he has been with his football for a long time," manager Greg Abbott told the club website.
"He's settled well with us this summer and I think he likes the fact that he's one of the senior players in the group.
"He thinks he can help these lads and when you see him chasing after causes, it makes it hard for others not to follow that lead. That's a great asset for us to have at our disposal.
"We have different types of forward now and that gives us options. We can play with two up top, we can go with a three or we can push our wide men further in.
"There's some pace and some real penetration there, so things are looking better than they were for us even just a week ago."