Switching bars affects bike fit. So, you may not fit well to a bike when switching handle bar types, since the way you hold onto the bars changes - you may need a different stem length (and height) or more saddle adjustment than on your current seat post than is possible to get a good fit (if its even possible - it isn't always). Note that well adjusted drop bars actually let you use the drops, not just the hoods or the middle of the bar or whatever. Also, drop bars have a different diameter where the brakes what not clamp on, so you need new brake levers.
Drop bars give you multiple hand positions (which can convert to more aggressive riding positions), which is useful on longer rides (20 km is a pretty short ride), so you won't really see much gain and you will have to buy new brake levers. So, I wouldn't recommend getting drop bars. With a flat/riser bar, there exist bar ends to provide a few more hand positions (Ergon GP3 is the set I use on my commuter, but primarly for the sweet grips rather than the ends).
Here is some good reading on setting handle bar height and types of handlebars.
Totally unrelated, but wow that saddle is tilted downwards.
Want more fitness? Ride longer and harder. Multiple speeds may help with hills (esp if you have knee issues).