WinLemm is notoriously annoying to get custom levels working with. It's not impossible if you're really determined (if that's the case, let us know again and we'll help you with the process, but be aware it's quite annoying), but it'll be much easier to use one of the unofficial Lemmings clones to get this working. I would recommend using NeoLemmix, but you may want to try them all out and see what your preference is (but from what I've seen, most people who initially prefer something other than NeoLemmix or Lix, eventually gravitate towards one of those two).
The viable engines (and whether you'd want to save your level as a LVL or INI file for them) are:
Lemmix (LVL): This one is quite nasty to set up by today's standards, but the advantage is it mimics the gameplay of the DOS version accurately, even including the bugs. If you do want to use it, let us know and we'll give more info, I don't want to type out a setting up tutorial unless it's the one you want to use.
Lemmini (INI): This is the one that jLevelBuilder is primarily intended for editing (other than LJLPM's own game which hasn't yet been released). Not too complicated, but is considered quite out-of-date. I think it has a "Load single level" feature somewhere.
NeoLemmix (LVL): NeoLemmix is the most popular these days. To play your level, all you have to do is run NeoLemmix, you'll get a popup box asking you to select a level or level pack to play; find your LVL file and select it, and there you go.
SuperLemmini (INI): This is similar to Lemmini, but I know for sure SuperLemmini has a "load single level" option.
Lix (LVL): Lix is the second most popular (after NeoLemmix) these days, it's worth noting that these days it and NeoLemmix probably have more in common than they have that sets them apart, with the engine of choice coming more down to preference and content than anything else - in a way, you can think of them as the PS4 and Xbox One of Lemmings engines, including that many people just use both. I don't know anything about how to get a LVL file to load in Lix, apart from that it's possible (native Lix levels use a custom format that jLevelBuilder does not support); I'm sure someone else will come along soon enough and fill in this gap, though. In general, the author of Lix discourages use of official Lemmings graphics for making levels (by comparison, NeoLemmix and SuperLemmini are fine with it), so if Lix is the one of most interest to you, you might want to consider working with the made-for-Lix graphics instead.
It might also be worth looking into moving to one of the more popular choices (NeoLemmix or Lix, or possibly even SuperLemmini) and getting the most appropriate editor for them, but we'll leave that more in-depth discussion for another time - if you want help with getting into editing for any of these other engines, just start a topic on the respective engine's board and we'll be only too happy to help you.
For now, let me know if this info helps get you playing your level, and if not, feel free to ask for further help.