Hi again Manoxlon - thank you for clarifying the situation regarding DD and her piano 'performance'!
Can you clarify a bit further: are you starting from scratch yourself, or had you played a bit previously?
Is it an acoustic piano, or an electronic/digital one?
And, yes musical 'games' are an excellent way of getting a young child involved. Depending on your level of proficiency, there are probably tunes that would be suitable for YOU to play, and for DD to add a contribution. "Pop Goes the Weasel" springs to mind, and she could either have a prescribed note (or group of notes for the POP), or if she has a little drum or tambourine that could be used.
There must also be 3/4 waltzes that could be suitable for DD to add her part to (though I can't think of examples off hand!)
I have looked at some people's Suzuki vs Traditional evaluations, and most that I saw favoured Traditional. But you could always buy the first Suzuki book, and adapt it to your own requirements. Also, I like the John Thompson series of books, his "Easiest Piano Course" or his "Teaching Little Fingers to Play", rather than the Usborne Sam Taplin one, which seems to use colour-coding, which I don't like.
The main thing however, is that you both ENJOY learning and playing.