I believe the spelling is "Linn" based on the following.
The Abstract of Early Kentucky Wills and Inventories by Estelle King p.130:
Linn, William. July 8, 1780. April 3, 1781. Wife: Letitia. Daus: Ann, Rachel, Theodatia, Lannie(?). Sons: William, Asabel, Ben. John and Josey Linn children of Rachel. If daus. Theodotia and Lannie(?) return from the Indians they are to have one thousand acres of land, but if they do not return the land to be divided among my sons. Friends James and Samuel Kibbie, John and Ben Earkinson to have 1200 acres below the mouth of the Miama. Wit: Charles Locke, Sanford Edwards.
The microfilm of Will Book A Jefferson County can be found in the Frankfort Ky, Archives Research Room. Some of it is barely legible but one phrase is very clear--"If they return from the Indians".
Also, from "The First Census of Kentucky", reconstructed based on tax records "p.59 Linn, William 6/18/1790" Jefferson County. This is probably either the father or the son; the date gives the date of the tax record.