The Fall and Rise of the House of the Wizard Malkuril by Scott Lynch. I read it in the anthology The Book of Magic edited by Gardner Dozois, and I'd guess this is the anthology you remember since the story apparently hasn't been collected anywhere else.
It starts:
The wizard Malkuril led a quiet early life, as materially successful wizards will, for one cannot otherwise survive long enough to amass the power required to live loudly. Meek and temperate were his first few centuries, but by the age of five hundred he was sleeping inside the fire of stellar coronas, and there were entire continents missing on certain planets to mark the occasional frayings of his temper.
Then we have Malkuril meeting his end:
He acquired the treasures of a thousand worlds, several million volumes of sorcerous lore, and the most comfortable pair of slippers he had ever owned, with a loose fit that was just right for the wide splay of his crooked toes. He wore them constantly at home and rarely bothered to lace them. In his eight hundred and nineteenth year, in his private apartments, the wizard Malkuril slipped on an untied lace and tumbled down a flight of thirty-seven stone stairs. The first twenty were merely painful and the last sixteen were entirely superfluous, as it was the twenty-first that broke his neck and killed him instantly.
As you say, his faithful Kobold servant continues to bring tea to his dead master:
“Tea for Master?”
The kobold peered at the scarlet-robed form sprawled at the foot of the executive staircase, blinked, and waited. Master took tea at the eighth morning chime. Fetchwell son of Fetchwell, luckiest, most honored of the high house kobolds, always poured the tea into Master’s favorite cup, carved from the polished kidney stone of a dragon. Then Fetchwell carried the teacup on a tray of pure iridium—(once, Fetchwell had thought that was all one word, puriridium, and been very proud of himself until the housemind found out and corrected him, banished him for months to the low house kobolds, banished him to work the wine cellar, and took all the luck and most-honor away, so now Fetchwell was careful to split the words right when speaking)—on a tray of pure PAUSE FOR BREATH iridium. So respectable, luckiest, most honored Fetchwell, so debonair (he never said that word out loud; he did not want to see the wine cellar again) for Master.
Here was Master, here was Fetchwell, here was tea, but Master was not taking it. Master was not even moving.
The sentient tower is Housemind.
