Condolences.
It's nice that the bike will continue.
I've seen several non drop bars that have had STI's fitted, mainly either bull bars or mustache type bars, I haven't ridden any of them, though the hand positions always look a little awkward to me.
You could have a go, though I can see how the costs of trial and error could easily be more expensive than simply swapping to what you know works. You can buy Shimano Altus 3 x 8 shifters for under £15. The only compatibility issue would be the front mech, though a replacement won't be dear. I don't know 9 speed prices but I don't think they're much more. You would of course also need new brake levers, I'd swap the canti's for V brakes at the same time, IMO that's usually a worthwhile upgrade and again not expensive. You could offset that cost by selling the STI's, some models fetch a good price.
Whatever you do, I hope you enjoy the bike.