The 1945 Hollywood, CA film that features Gene Kelly dancing with an animated mouse is "Anchors Aweigh". In addition to a live-action Kelly dancing, Jerry Mouse, an animated cartoon mouse, is used as Kelly's dance partner.

In its entirety, the film "Anchors Aweigh" has Joe (Gene Kelly) and Clarence (Frank Sinatra) as 2 young U.S.Navy sailors on leave in glitzy Los Angeles. The 2 men at first plan to use their layover as an opportunity to relax, but they soon find themselves in a complicated web of comic secrets. When the two buddies find a runaway boy and return him to his fame-hungry Aunt Susan (Kathryn Grayson), both men find themselves falling for the wannabe singer. This will set off a series of misadventures that unfold in comedy and song.

In a scene, Joe ends up telling some kids the story of how he got a military medal. He also points out how he brought (1) happiness to a lonesome king (played by Jerry Mouse from 'Tom and Jerry', with Tom briefly appearing as a servant), and (2) joy to animals in the forest in the kingdom.

"Anchors Aweigh" is a musical comedy that was directed by George Sidney and produced by Joe Pasternak. The screenplay was written by Natalie Marcin and Isobel Lennart. Major stars include: Frank Sinatra, Kathryn Grayson, and Gene Kelly,. Songs were performed by Jule Styne and Sammy Cahn. It got 5 Oscar nominations; it won the Academy Award (Oscar), Original Music Score, Georgie Stoll. After its release, a profit of $2.1 million was made.

More Info: en.wikipedia.org