how to get an historical family members records Omagh Asylum

Hello all, new member here, located in Australia.

I'm hoping for a little help in how to track some information on a newly found member of my family who died in 1909. I thought I had completed this branch but just stumbled across this new long lost relative.

His name was Bernard McEntee, born circa 1845. He died age 63 in Omagh Asylum on 16th January 1908. The only reason I know he must be a brother of my great grandfather is the death registration shows (under his name) he was from Eshywulligan (although a little difficult to read), Roslea, County Fermanagh. The McEntees are a rarity in Fermanagh and there was only the one family in Eshywulligan (1901 and 1911 census). In the 1901 census the townland was shown as "Eshawilligan"

Is it possible and how would I go about accessing his asylum records ---- I am hoping they may shed a little more light on the family I have attached his death registration.

Thanks in advance for any help members may be able to give.

Kind regards,

Bill