I'm currently making an album, and I wanted to include a cover version of 'I Can't Make You Love Me'. How do I go about obtaining copyright permission? Is there a quick/free way of doing this?

are you planning to sell the album? or make any financial gain of any description?
if so, then obviously you would not be able to obtain the rights for free (that would be crazy), and if not then thats another matter entirely. A written enquiry to the company in question would be a good start.