Rhythm Control 2 is a free to play rhythm game with a nice variety of songs from Japanese songs to English songs. If you're familiar with a free rhythm game on the PC called Osu or a DS game called Elite Beat Agents, you'll probably like like this game because it has similar game mechanics. Not just that, it's way better than the Osu game app itself if you played that app.
Anyway, this game had very cute graphics and also has a nice selection songs. Not much else to say since I'm not a hard core Rythm gamer but do check it out on the App Store!
Conclusion: Awesome rhythm game and best of all, it's free.
Note: Available in the US and Japanese App Stores. Leveling up in-game will allow you to unlock more free songs and content.