You've got a couple things going on. First, the boominess is probably from a peak in the sub's response caused by driver misalignment. You need exact calculations of enclosure volume, port diameter and port length for each specific driver. Unless you get REALLY lucky, the response of the sub is going to be suboptimal with boominess being characteristic.
Second, you won't be able to replace the stock crossover by simply adding an active one. If the stock one is passive, you'll have to build a new one specifically for your speakers/sub. If it's active (which it probably is) there is nothing you can do to change it. In the latter case you can use an external active crossover but you'll also need to buy an amp (like BoredSys stated).
The last thing is more an FYI. Power ratings on speakers are pretty much useless because they don't tell you the conditions under which they were obtained. Plus, power rating and impedance will only tell you if the speaker and/or amp will fail/blow up/etc if used together. They don't tell you anything about the sound quality of the resulting match. Using the Polk speakers isn't a problem (besides the crossover issue) but replacing the sub driver is for the reasons stated above.
The best solution would be to scrap/sell the remnants of the Logitech and build an enclosure for the JL driver. You'd need to get several amps and maybe a crossover if your computer doesn't have that function. Otherwise, the next best solution would be to put the original sub driver back in. The least ideal solution would be to try to equalize the sound to your satisfaction but that's a whole other can of worms. Please remember, the Logitech was designed to work as a system without a good way to upgrade individual components.