Get three taproots
Notice! This mission will be available only after you complete Path of Knowledge!
Speak to Elynea Mothren in her apothecary (the tower to the left of Tel Mithryn) and the woman will share her concerns with you. Master Veloth will fire her if the sick mushrooms aren't cured fast (after all, they are the mage's house). The only cure is a solution created from a certain very rare alchemic ingredient known as taproot. It can be obtained only from the best alchemists or... from the corpse of a deadly nymph Spriggan. These creatures however aren't very common.
The situation seems to be dire, but there is one place in which you can find exactly three taproots. You need to head to the furthest edge of Solstheim - to a small island in the north (a bit away from the Frossel cave). You can find as many as three Spriggans there - fighting them can be quite problematic, but it's surely easier than searching for the rare ingredient on your own.
Collecting the roots is only the beginning of the mission - you need to prepare them properly, which will be the task in the further part of this mission.
Soak the Taproots
The taproots are useless on their own - for them to have the proper power, you have to soak them in a sacred spring. There is only one such place in Solstheim - Headwaters of Harstrad. This place can be found west of Wind Stone and southeast of Benkongerike (screen above).
Head there and kill another Spriggan who's guarding the area - defeat her and approach the waterfall. Go through the wall of water and head inside the cave behind it. Inside you will come across a deadly Spriggan Earth Mother, who has lots of health points and a very powerful attack. In fact you don't need to fight her - just soak the taproot in the water (you do it by "using" the river).
At the back of the cave you can find a treasure chest - in order to reach it, you will have to defeat the guard. She's very vulnerable to fire attacks and resistant to physical damage, which should be a big hint for you.
Talk to Elynea
Return to Tel Mithryn and meet with the alchemist. The woman will be happy to accept the soaked taproots from you - she will keep two for herself and give back one to you. Use the root and plant it in the sick tree. That way you will cure the whole construction and everybody will be able to sleep well. Therefore you will also save poor Elynea from being fired.
Plant the taproot
Head out of the apothecary and into the main tower of Tel Mithryn (the one in which master Veloth works). Use the magic teleport which will take you upstairs and turn right. On the wall you should see a giant, swollen abscess. Use the soaked taproot on it and wait for the effects. Afterwards you will be able to return to Elynea and give her the happy news - the mushroom has been cured.
Talk to Elynea
Return to the apothecary and tell Elynea that everything's done. In return you will receive two Potions of Well-Being which restore health, magicka and stamina. The quality of the potion depends on the level of your character - the higher it is, the more powerful the potion is.
At this point the mission ends - who would have thought that renovating your house might be so problematic?