Westlake had previously been on loan at Whaddon Road.
"Ian Westlake signed a contract yesterday (Friday), so I'm delighted about that," boss Martin Allen told the club's official website.
"He's a good player and a good person and he did well to play today (against Brighton), because he's been injured.
"It's great news for Cheltenham Town that he has terminated his contract with Leeds United and he remains with us until the end of the season."
The former Ipswich Town player joined Leeds in 2006 and made 52 appearances for the club, but his last outing came 12 months ago and he did not figure in the plans of new manager Simon Grayson.