If you ever decide to do a full Hanukkah mixtape (or several), don't restrict yourself to the known knowns. Like Christmas, the themes of the holiday can be expanded beyond the obvious. One place few ever think to look to is Israel. There are a great many contemporary artists in Israel creating a variety of wonderful sounds. Their music may be Rock or Jazz or Folk or Hip Hop, but it usually also contains a good measure of traditional influence reflecting the people and culture. Songs that deal with themes of love and perseverance may not technically be Hanukkah songs, but fit the broader context of Hanukkah (think "Put A Little Love In Your Heart" from the film "Scrooged" as your Christmas analogy) and will make your mixtape sparkle. Just the music, Stubby.
One of my favorite Israeli bands is Moshav. Back in 2006, I used their song "Don't Give Up" on my "Stubby's Chanukah 4: A Chicken Suit Is Good For The Soul" mix. Not a Hanukkah song, but an excellent Alt Rock tune well within the themes of the Hanukkah story.