In a 1846 commercial directory under the listing of watch and clock maker.
Henry Doyle (clock) 43 Mill St. Belfast
No other information:www.emeraldancestors.com
free to view but pay for records
Also for Co. Antrim is an Irish History Foundation that
has parish records, a birth record for Emily should give her mother's name. You have a good birth year and you can search for Emily with father Henry (in all Ireland) or just Antrim. This website is a not for profit History Foundation but fee's are charged.
Looking at the Parish records there aren't a lot that go back far enough to get a birth record for Henry but you might be lucky with a marriage record.