I agree that you'll probably eventually want both (no one said that this is a cheap hobby!), but I'd go with the flash first.
It's easier to use quickly, and works well when you have multiple subjects not necessarily side-by-side — or when your kid won't stay within the in-focus area for for than a millisecond. With indoor lighting, even a wide-open fast prime is sometimes not enough.
Nikon has a nice, simple, and affordable flash with bounce capability, the SB-400. They make a lot of great flashes, in fact, and you certainly wouldn't go wrong getting a higher-up model, but you'll also get a great improvement just from the very basic model. Plus, it's small, which means you can include it in a small camera bag without much sacrifice.
Ooh, one more thing: a flash lets you take fun pictures of babies knocking over towers of blocks, with the blocks frozen mid-air:
Now, admittedly, that's with a nice prime and a nice flash, but the prime is set to f/5, which you could easily do with a low-cost zoom.
This one was taken of my other daughter, using an Olympus "bridge" style camera with a small sensor and built-in zoom lens. Your DSLR is unquestionably much more capable even with the kit lens. I added Olympus's low-end hotshoe flash, which is roughly equivalent to the SB-400. There's a bounce card which gives the catchlights in the eyes, and the flash is otherwise pointed straight up. This doesn't give very exciting, dynamic-feeling lighting, but it looks pretty nice. I've got some examples of more dramatic lighting, but the point here really is that a flash can make meaningful improvements even used very simply.