Replace crankset or cassette for better road performance on an MTB?
Since you already have an 11 tooth smallest cog you're not going to be able to swap that for a smaller on. Well, you can get a 10T cassette but I don't recommend it, they're designed for small wheel bikes where the faster wear on that cog is just part of the deal.

Swapping to a road triple crankset should be do-able, and with any luck your front derailleur will work with the new one. A cheapish 30-39-50 is about $60 on ebay or amazon, so probably a bit over $US100 fitted if you go to a bike shop. But since you'll need extra chain to wrap around the bigger chainrings, best budget for a new chain and rear cassette (normally those get replaced together. See here and here for why) which will probably be another $100 or so. Call it $250, but ask for a quote because this is a pretty standard swap. 

If you're unlucky you might need a different length bottom bracket ($50-ish), or a new front derailleur (plus gear cable, $100-ish). Those are things the bike shop probably won't know until they start fitting parts to your bike, so the only way you'll get a certain quote up front is if they decide to replace them regardless.

That will give you noticeably higher gears across the board, but obviously you lose the low gears (you'll effectively lose your smallest chainring and gain a much bigger one at the other end.

Or fit a slightly larger MTB triple. That will give you less gain in range but is more likely to work with your current derailleur and bottom bracket. You can easily get 28/38/48 sets, and the difference in the top gear between 48T and 50T is small. But that's the problem - you're going from 44 to 48 or 50T, and while either will be noticeable, it's not as much as you might think.

The difference is in how easy it is to go even bigger. On a road triple getting a 52/54T chainring is easy, but with an MTB crankset those chainrings get pricey if you can find them at all. We have a local manufacturer who CNC cuts chainrings, so we can get anything we like. I expect people like that are on the web as well, but you'd have to look (and try to buy local since you're fairly likely to want to swap them when they don't work on your bike because they're too big).

A completely different approach would be to fit fat slick tyres. Or even fat knobbly ones. If you're riding for fitness, the extra drag (and very slightly higher gearing) will be just what you want. Schwalbe make their "fat frank" tyres as big as your bike will take, or "Big Apple" out to 2.5". The big tyres will be noticeably harder to push along at the same speed. And remember, as the cops here keep telling us "the faster you go, the bigger the mess". Going fast just because you can, rather than because you have somewhere to go, doesn't make a lot of sense. I commute ~20km each way on a nice comfortable upright commuter bike that will go 30kph if I push. So I get fitter faster for the same distance ridden. And when I'm lazy I pull out my drop bar racer and do the same ride in 3/4 of the time (33% faster average speed). But I'm more likely to crash on that bike, and if I do it'll hurt more.