I did a quick check, and found him in the Chicago census for 1870 and 1880. Do you know where he was in 1860? Since most immigrants, including the Manx, tended to live with others of their nationality, if you can find him in the 1860 census, and then search the pages before and after his entry, you might well find a family with a daughter Louisa.

Have you tried looking at the Illinois Marriage index

http://www.ilsos.gov/GenealogyMWeb/MarriageSearchServlet

I would look, not only for Thomas' marriage, but also for his children's (because they would name Thomas' wife). You will have to be creative about the spelling of the name Quayle when you search indexes: try Quail, Quale, Quayl, etc. Use the Soundex option if there is one.

Any clues about what church they might have belonged to, or where/when they might be buried? If you can find a date of death, and then an obituary, that might give you more information. There is an Illinois death index, but I am not sure it is complete. You would get to it from the same place as the marriage index.

Unless he immigrated with Louisa, it will be hard to find his immigration record. There will be, doubtless, individual Thomas Quayles, but it is very hard to identify an individual as "the" one. (The route he took was most likely Liverpool to New York)