Your mini lux most likely has 14.5mm joints. However, you should try and look it up online just in case, so you can make sure you get a downstem that fits. As for the length, try and measure the length of your original diffy (or estimate how long it was before it was broken if you can't get a close to exat measurement). Remember that it's okay if you get something that's slightly too short, because it'll just mean you'll have to use more water in your bong. However, if you get a downstem that's too long it won't fit securely into your bong.
Make sure you know the return policy at your headshop in case you get a diffuser that doesn't fit. Also, take this opportunity to try and fit your lux with some extra shit, like maybe an ashcatcher or a sweet bowl.