In hammer action boards, I think all the suggestions here are overkill, going to the higher end of OP's budget unnecessarily. Since no sounds are required other than piano, with budget a priority, I'd look at Yamaha P-45, Roland FP-10, and Korg B2, all under $500. which I think are will sound and feel sufficient for these purposes. The Korg starting price is a bit higher than the others, but if you want an aesthetically nice stand, a pedal that is more like a real piano pedal (instead of a small square footswitch) and bench, the three will all end up in the 6-something range, I believe. The Korg has the nicer pedal out of the box, but also, gets you an optional triple pedal, if she might want that... the B2SP is the B2 bundled with the stand and triple pedal, and it's probably the cheapest model with that functionality.
Personally, I'd probably lean toward the Korg. I have the step-up XE20, and the action is particularly quiet, which might be beneficial here, for not wanting to bother others in the house when playing with headphones, and while sound is always somewhat subjective, I think it also might beat the others in sound. I happen to like Korg's Italian Grand sound quite a bit, and while the B2's speaker/amp system is a step below the XE20, it probably still beats those others, it certainly does on paper. At any rate, I'm not sure I see any reason to go beyond these in price unless something about these are unsatisfactory.
Happyrat's idea of going to a semi-weighted may also be worth considering... it will feel less like a real piano, but will be easier on her fingers, if that might be turning into an issue at her age. But while the Numa Compact 2X--or the lower priced non-X version, which lacks some of the organ, synth, and MIDI functionality you presumably don't care about anyway--would give you a lot of features and capabilities for the money, keeping in mind that the only function here that matters is piano, I think the less expensive Korg Liano sounds and feels much better. Some of that is that the speakers are noticeably better, some is that control of dynamics is better (perhaps related to the fact that the keys don't push back against your fingers so much, and they feel more even in resistance between the front and rear of the keys). I do like the Numa series for all they offer, but strictly as a piano replacement, I think you'd be paying more for less.
Still, even if finger strength is becoming an issue at her age, I wouldn't rule out the more piano-feeling hammer actions I mentioned above, though it might be another reason to lean toward the B2 which I think is the lightest/softest feeling of the three (with the Roland probably being the heaviest feeling).
