I used those same speakers for 2+ years and was pretty happy with them, too. When it came time to upgrade, I discovered that a significant improvement was going to cost a lot. In my case, the replacement speakers were $2600. Definitely a step up, but the improvement-to-price increase was not linear.
Figure on spending a minimum of $1500 (likely more) to replace your current speakers, versus $300-500 for a competent subwoofer.
Not only will adding a sub save you money, it may even outperform a more expensive system that doesn't need a sub. That's because speaker placement is critical for low frequencies, and having them coming out of a separate box means you can experiment with moving them around the room to find the best spot. If, OTOH, you get high-end full-range speakers that don't need a sub to fill out the low end, their placement is predestined with little room for adjustment.
IIRC, your Alesis speakers should be reasonably flat to ~60Hz or so. That means a sub will need to handle less than two octaves, meaning you don't have to spend a lot of money. As long it's capable from 30 Hz to 70 Hz it'll positively contribute to your mixes. Something like
this will fill the bill on the cheap. (Yes, you can spend a
whole lot more, but 2 grand on a sub is not a great ROI.)
So now that I've sold you on the sub, I'll try to talk you out of it.
Not hearing frequencies below 60 Hz isn't necessarily a bad thing. You can still check for excessive or insufficient loudness in that range using visual aids; believe it or not, for most genres it isn't technically necessary to actually hear them. (The possible exception being EDM, which typically features a lot of very low frequency content and is often played back on systems specifically designed to reproduce them.)
Not having frequencies below 60 Hz means they also can't cause problems. And the biggest problems in any monitoring environment are down there. In most rooms, there will be resonances in the first two octaves that you will not be able to easily mitigate with acoustical treatments. Unless you're prepared to seriously study the science of acoustics and/or hire said expertise, you're probably better off just staying out of those bottom octaves altogether.
Now, all of the above is predicated on the assumption that money is a concern. But if money is no object, forget everything I just said. Take about 40 grand, split between
full-range mains,
dual subs and an acoustician.