W H Neale states that Lieut Monk, R.N., was the first of the well known Monk family in Ramsey (see www.manxnotebook.com/towns/ramsey/whn_07.htm) - I presume you have looked at Constance Radcliffe's books on Ramsey ? - I think it was this Monk that built the windmill (Lezayre - completed 29 August 1836 [ref Mona's Herald] apparently built by Lieutenant John Monk who employed millers to run the mill ).

There may well be some info at the PRO as he was in the Navy

There was also a Lieut Monk in the Northern Manks Volunteers c. 1798 (see www.manxnotebook.com/iomnhas/lm4p096.htm) but at Michael - there may well be additional info here, not clear if same family