Did you buy Danwguard then try to buy Breezehome? The add-on has a glitch that leaves Breezehome's alchemy room completely empty and you can't furnish it.
Yes, I bought dawnguard, I suspected it was the mod since it's the only thing who affect breezehome directly.
Anyway this should be possible to fix with an patch or command line entry on pc. as it breezehome work if you expanded it before dawnguard.
Found some console comands who is said to solve it, will try then back from work.
Thanks for nailing it down.