The dragon soul bug can be one of two issues.

There's a bug in vanilla Skyrim where dragons that have respawned don't burn up and give a soul. The Unofficial Skyrim Patch should fix this. However double check by killing another dragon elsewhere (preferably a randomly spawned one) to see if it isn't this problem.

There is also the more common problem where you can't absorb any souls. This is either due to an incompatibility issue between certain mods and the Dragonborn DLC, or a flatout broken mod.
If you google about this problem you will find people complaining that the USKP causes this. Depending on the mods they're using and their loadorder they can - purely technically - be correct. However that's a bit like holding a bullet responsible for a murder, instead of the guy who pulled the trigger.
Secondly you can find a lot of people telling you to uninstal the USKP from your playthrough. This is really, [i]really[/i] bad advice. Do not deactivate the unofficial skyrim patch and then continue to play with an existing save. In Skyrim's Creation Engine you cannot freely remove script-heavy mods (like the USKP) from a save like you could in the previous Gamebryo Engine. If you do this, you will damage your savefile and end up with issues such as savegame bloat.
A mod creator can setup an uninstal procedure to remove their scripts as much as possible from your save. But if there's no such uninstal procedure and you want to remove a mod that uses scripts, either start a new game or revert back to an old save from before you installed the mod.

Now to fix the problem:
First make sure you have all of the unofficial patches (excluding those for the DLC you don't have).
Secondly make sure you have a proper loadorder. If you're using BOSS it can sort this out for you. If you don't use BOSS or want to check its work, your loadorder should follow this template:
Skyrim.esm
Update.esm
Dawnguard.esm
Hearthfires.esm
Dragonborn.esm
<high res texture pack plugins & .esm files added by mods>
Unofficial Skyrim Patch
Unofficial Dawnguard Patch
Unofficial Hearthfire Patch
Unofficial Dragonborn Patch
<all other plugins>

Note that it's possible that the unofficial patches will switch to a fake .esm format at some point in the future. When that happens, the loadorder template above will likely change.
If you're using the Skyrim Launcher to set your loadorder, it won't show Skyrim.esm & Update.esm. Don't worry, it will load those two first automatically.
If you're using Mod Organizer, be sure that your mods are installed in the right order. That's the left side of the screen in MO. General rule of thumb: the order of your installed mods should be the same as the loadorder of your plugins. Also be sure that you've activated the patches' bsas under the BSA tab. When you do this, pay attention to the order of the patches' bsas. If they're not the same as the above template, go fix your installed mods order.

Now to hunt for any remaining possible problem sources.
If you're using Mod Organizer, start it up and open the "Data" tab. This shows the virtual data folder MO will voodoo into existance and which mod files belong to. We're going to check for incompatibilities and improper loadorder. Scroll down and expand the "scripts" folder.
First look for "dragonactorscript.pex". It should be the version from the Unofficial Dragonborn patch's BSA. If you have the version from the Skyrim Patch, you messed up your loadorder and/or installation order. Go fix that, then reboot MO and check again. If it's supplied by another mod lower in your loadorder, asume for the moment that this mod is incompatible with Dragonborn and remove it from your loadorder.
Next look for "mqkilldragonscript.pex". If you do not see this script, you're good. If you do find this script, you're not good. If it's located under "data", that means you probably once upon a time installed a mod that messed with the dragon souls absorbtion animations & VFX and either forgot about it or didn't uninstal it properly. Find the file under /skyrim/data/scripts/ and delete it. If it's supplied by another mod lower in your loadorder, again assume for the moment that this mod is incompatible with Dragonborn or broken and remove it from your loadorder.

If you're not using Mod Organizer, open /Skyrim/data/scripts/ and look for "dragonactorscript.pex" and "mqkilldragonscript.pex". If you find either of these files, delete them. If they're present, they can be either loose files that are left over from a mod that you didn't uninstal properly. Or they're loose files belonging to a mod that you currently have installed. Find out which mod added them and uninstal that mod if you haven't done so already.
These two scripts can also be packaged inside a .bsa. Use a tool that allows you to view the contents of bsa archives (for example [url=http://skyrim.nexusmods.com/mods/247]BSAOpt[/url]) and take look inside the bsa files you're using. If "dragonactorscript.pex" or "mqkilldragonscript.pex" is present in any of the archives (besides those belonging to the unofficial patches), remove that mod.

Once you've done all that, go test if the problem has been resolved.
Note that that - due to how Skyrim's save system works regarding scripts - even if you fix the problem, it's highly likely you won't see any change when you load up a save directly before or during the dragon fight. Load a save before the fight and where the dragon you're fighting isn't in the same cell as your character. If you've ended up removing mods during this process, it can be best to just start a new game and quickly race to the Mirmulnir fight on godmode (use the "tgm" console command to activate and deactivate godmode).

If you cannot absord a dragon soul even on a new game, go start again from the beginning of this post and double check every step.
If you can succesfully absord souls congratulations for being dovahkiin again. If you want to instal any mod that affect dragons, be really sure that the mod in question works properly and is compatible with Dragonborn. If there are any doubts, asume it is broken.