What was the largest empire in human history in terms of the time required to deliver a message from one end of the empire to the other? How long was this time?
In terms of the area of control established by communications, unquestionably the British Empire - with telegraph connections to Australia established in 1871, Britain had enormous swaths of the globe - from Tikitiki on the North Island of New Zealand all the way west to Beaver Creek in the Yukon, with stops for the Indian subcontinent and an enormous chunk of Africa along the way - under its direct control with end-to-end communication.
In terms of the length of time it took to reach the farthest corner of the empire, the British still win, with some sea routes to its farthest possessions taking many months even with the sophisticated and well practiced maritime practices of the 19th century. It was much longer with older vessels and navigation techniques.
It is a strange way, of course to measure the SIZE of an empire in terms of the TIME of travel from end to end; at different times different means of transportation were available, and it also depended on the season.
But if one accepts this strange criterion, perhaps the Russian empire in the early 19th century is hard to beat: it stretched from Alaska and California to Poland. The only ways to travel from the Far East to the capital was either around the whole Asia, Africa and Europe by sea, or by sledges across Siberia. The land travel from the Far East to St.Petersburg took several months in the best case, usually about 6 months, but sometimes more than a year. Traveling the "other way" (not through the Empire but through America and the Atlantic Ocean) was in fact sometimes faster, because ships travel faster than sledges.
The Mongolian Empire at its largest extent was almost the same size, excluding Alaska and California. It stretched from Korea to Poland, but they probably had better postal service at that time. The speed on land widely varied: it depended on the infrastructure and on the ability to change horses and to buy supplies. A traveler with much money or on an official mission would easier obtain horses and supplies. It also depended on the season: in summer one had many rivers to cross (including very large ones), while in winter they can be easily crossed by sledges on ice.
When Batu and Subutai conquered Poland, the Great Khan Ögedei died in Karakorum. (This was the capital of the empire, about 2/3 of the way from Poland to Korea). He died in December 1241. Batu and Subutai left Poland in the "late spring" 1242. I suppose they left immediately after receiving the news, and that the news was passed in the fastest possible way).
EDIT. On October 7 1852 a Russian diplomatic mission started from StPetersburg to Japan (by sea). Japan has a common boundary with the Russian far East possessions. They arrived to Japan in the beginning of August 1853. So it took 10 months by sea. The mission returned by land. They started on August 15 1854 in the Russian far East, and arrived to StPetersburg on 25 of Februaty 1855. 6 months by land. The travel is described by the secretary of the mission, the famous Russian writer Goncharov in his book Frigate "Pallada".
In 1904/5, during the Russian Japanese war, the Russians sent a fleet (consisting of steam ships) from the Baltic to the far East to fight the Japanese. The fleet departed on 15 October 1904 and arrived to Tsushima straight in Japan on 28 May 1905. At that time a railroad crossing the empire from West to East already existed. So we see that steamships in great hurry made this in 7 months while a sailing ship (perhaps not in so great hurry) made it 50 years earlier in 10 months.
The answer for land empires may have been the Mongol Empire due to its enormous size. (After seeing @RISwampYankee's answer, he probably has the correct answer over all.) However, the Mongols were not only incredible horsemen but also had a very efficient postal system called the Yam.
According to National Geographic:
At the postal route's zenith, a letter could cross from Kharkhorin in the east to the Caspian Sea on the far western edge of the empire, a distance of some 4,225 miles (6,800 kilometers), in two weeks (an average of about 300 miles, or 480 kilometers, a day). Postal riders continued to deliver the mail until 1949, when the Soviet Union—which then controlled Mongolia—shut down the system in an attempt to erase the history of Genghis Khan from the country.
Of course, the time it would take a message to get from one remote part of the empire to another equally remote part could be considerably longer than two weeks.