Need budget replacement for broken kit lens - am I missing something?

I have read the other responses to your questions, and I am giving you a middle of the road opinion.

Buy the 35mm 1.8 if you also want to explore photography. You will learn new things, get a new experience with getting background out of focus, which is nice for indoor kid's pictures. Remember that 35mm on D60 will give you a effective FOV of almost 50-55mm, so framing could be tight.

With this you will obviously miss the other focal lengths, mostly the wide end. For that you can stick to your point and shoot for now. I agree with the posters that said that it will make you more creative, even though its limiting. For wide landscapes outdoors, you could stitch using something like Hugin. You will lose shots which require longer focal lengths.

Eventually you will need the wider and longer focal lengths, so start saving or looking for a good deal on 18-xx kit lens. 18-55 kit lenses are about 60-70$ on keh.com, if you stretch your budget a little bit you can have both.

I would also consider the 50mm f 1.8 AFS, which is about 220$, but a more classic portrait length, but will be more difficult to frame with indoors.