According to this article on Bikeradar, "Current 9-speed road components are backwards compatible with older 9-speed road and mountain bike components, excluding the aforementioned exception" as well as "Current 9-speed Shimano mountain bike components are compatible with older 9-speed road and mountain bike components, excluding the aforementioned exception". In this case, the exception is pre-1997 Dura Ace componentry, so that shouldn't be a concern for you. This means that you should have no compatibility issues at all regarding shifting. The pull on Shimano road levers will be fine for the Tektro Novela brakes as well.
While you won't have any compatibility trouble just swapping the shifters over, you may find that the gearing is far from ideal for a road bike. 44-11 is a very low top gear for a road bike, so you might find that you want to swap the crankset with two chainrings as well. You will still be able to use your rear derailleur, but this will mean that you need to add a front derailleur to your bike and a new chain as youlr currrent one will be too short as well.