An act of the NH state legislature settled the spelling of the name of the Lake in 1933. Article below from the Laconia Citizen newspaper.

Here are some of the names for the Lake seen on historical maps of New Hampshire, and the date when a particular spelling is first observed. The spelling “Winnipiseogee” was most common during the middle of the 19th century, having appeared first on an 1816 map of New Hampshire by Phillip Carrigain. The spelling “Winnipesaukee” gained favor beginning in 1885 when it appeared on a map of the Lake published by the Boston & Lowell railroad. Many of the other spellings may have been accidental misspellings and were probably never in common usage.