Stuff for kids is not as simple as a lot of people first imagine it to be, grammar being a particular problem, and translation is not always a good way to learn (e.g. the bit where you try to get everything into understandable English). However, I do think games are a good fun way to practice - not if you're trying to translate every text box, but if you understand the instructions and can work out how to play the game, that's always a plus! As long as it's in addition to study, not instead of it.
If you have a DS, the Zelda game Phantom Hourglass or whatever it was has pop-up kana (tap the kanji), and was pretty fun. I haven't played anything else that was all-kana, but I know some stuff is. I do like online RPGs or city-building type things, since you can work your way through the tutorials and it's pretty much along the lines of "go here, kill three wolves, then talk to so-and-so" or "click here and equip your new helmet" or whatever. There's loads of stuff like that for free on places like bg-time.jp/ or
無料ゲーム・オンラインゲーム・PC ゲームのハンゲーム but none that I know of that's kana-only.
Other stuff that is online and free:
Hiragana reading/writing practice sites:
勉強しよう!Leeのきょうざいかん・国語・こくご・ひらがな
(All the various links underneath おぼえる are basically "match the kana word to the right picture" tasks with various themes)
ひらがな 練習 プリント [ジャッカ -JAKKA.JP-]
(More for writing, meant to be traced over or copied, but any link here which contains the word "短文" gives short simple sentences in hiragana so could be used for reading practice).
Major kids portal sites:
kids.yahoo.co.jp
kids.goo.ne.jp/
These will have a few simple online games/study 'games' and a variety of other stuff. Some is kana-only, other stuff will only use low-level kanji. At least see how much of the front page you can work out.
Free (legal, older+out of print stuff only, ad-supported) manga site, some of which has quite a bit of furigana, try a few (grammar/vocab is likely to be the issue):
Jコミ