There is an interesting set of records for William Insley at the Family Search database [https://www.familysearch.org], and in reading between the lines I can see a possible reason for your difficulty in finding his family.. If you create a (free) account there, you can find William in two separate records, under ID ## M3M6-NN8 and LBR3-FMD, which appear to be duplicates created by the same person If you go to either of those records, you will see a link to click on to get to the other.
The records refer to William Insley aka Patrick Ryan in one case, and the reverse in the other case, and mention his death in New Zealand in 1881. I don't see any explanation given for the alternate names, but in each case his birth parents are shown as Patrick Ryan and Catherine Woods. If you move to Catherine's record (through a link in William's record), it appears that she may have remarried in Hobart, Tasmania, in 1850, to a William Insley, at a time when her son Patrick would still have been a young child. The possibility which occurs to me is that he was adopted by his stepfather and was renamed for him, so Patrick Ryan became William Insley. I have no idea whether that is true, and if so the person who created the records did so incorrectly (there are better ways of referencing a name change or adoption in an FS record, rather than creating a competing record).
When a record is created or modified in the database, the screen name of the person doing so is indicated to the right within the record, and you're able to send a private message to that person to start a conversation. You might do that and compare notes with her (Christine Walton seems to be the person in this case). She should be able to explain the situation leading to the two names for the same person.
The record for Catherine Woods also gives the names of her parents and some siblings, and says that she was from Annakissy (in Irish, Áth na Cise, presumably referring to a ford made using crossed twigs). That is the name of a Catholic parish in Cork, to the east of Mallow. You can find the parish records for Annakissy online at this link:
https://registers.nli.ie/parishes/0006
Those records go back to 1805 for marriages and 1806 for baptisms, so you should be able to find info there about Catherine and her family, as well as the Ryan's, if they were from the same parish.:
The reference to Annakissy may simply refer to the Catholic parish, but there are also three adjoining townlands in Cork with that name: Annakisha, Annakisha North, and Annakisha South. The latter two are in the civil parish of Clenor, and the first one is in the neighboring civil parish of Carrigleamleary, which all appear to be within Annakissy Catholic parish (though it's a bit hard to be sure from the available maps). The civil parishes are administrative districts, rather than church parishes). You can find more info about Annakisha North at this link, and use further links at that page to move on to the other two townlands:
https://www.townlands.ie/cork/fermoy/clenor/clenor/annakisha-north/
On the page for each townland, there are also links to records for that townland from the 1901 and 1911 censuses, as well as Grffith's Valuation (mid-1800's), if they are available.
If you find a baptismal record for someone in the parish records, it will normally also state the name of the townland where the family lived.