I think there are no brifters (combined brake levers / shifters) in 9-speed for hydraulic brakes (and couldn't find them either, same as you). The reason for this is that hydraulic for road rim brakes were never really a thing, and when the first hydraulic disc brakes appeared, gear count was already at 11.
if you really want to do this, I think the best way would be to change the rear derailleur and the cassette+chain as well to an 11x for which there are hydraulic brifters. Brifters are the expensive part of this refit (at least if you go for the more budget-friendly parts), an RD doesn't add that much, and the cassette+chain are consumables anyway.
If you want to keep your drivetrain, you could in principle buy brifters without using the shifting part and in addition bar-end shift levers, but it would probably be more expensive than replacing the read derailleur.
For left-side only there are also brake levers without shifting functions (ment for 1x drivetrains).
Another thing to consider is geometry: when you get a drop bar, your hand position on the hoods will be more forward than on a straight bar, probably also more forward than on most bar-end configurations.
Hand position on the drops is not comparable at all, of course.
A road bike that fits you well might have a shorter top tube to compensate (This is just an informed guess: you could check on comparison websites like 99spokes)
I think the best approach is to get another bike (and maybe trade in the old one). If you can get your hands on something used, this might be the same price and less effort. And I think it would be a better bike compared to your converted one.