The 22-year-old right-back joined the Nottinghamshire club, where he will remain until January 5, from Sheffield United on Thursday.
Westlake will provide cover for Ritchie Sutton, who has been ruled out for six weeks with a thigh injury.
"Darryl's just the kind of quality addition we're looking for," Cox told BBC Sport.
"First and foremost he's a good defender, particularly in one-on-one situations, but he also offers a lot going forward. When you've played at a higher level as he has, then dropping down to League Two, he can add that extra dimension and bit of quality."
Westlake has made 10 appearances for his parent club this season.