Isabella’s birth is long before the start of statutory birth registration in Ireland (1864). So to find a record of her or her family you need to rely on church records. Do you know what denomination she was? I looked in the 1901 Irish census, which is the oldest complete census) and there were 37 people in Armagh named Powell. 18 were RC and 19 were Church of Ireland (ie Anglican), so an even split. Those seem to be the 2 likeliest options.
I searched on the rootsireland site for Jane’s birth and her parents marriage but did not find either.
Not all churches have records for the early 1800s, and of those records that do exist, not all are on-line. The RC records are nearly all on-line but the Church of Ireland records are not. There are copies of most of the surviving Church of Ireland records in PRONI (the public record office) in Belfast but a personal visit is required to view them.
Looking at Griffiths Valuation of Ireland for Co Armagh in 1864, there were Powell households in just 2 parishes – Loughgall and Lisnadill. They would seem two obvious parishes to search. Loughgall Church of Ireland’s records start in 1706 (with some gaps) and the RC church in 1844. Lisnadill has no RC or Church of Ireland records earlier than 1835. (The Church of Ireland’s early records were lost in the 1922 fire and the RC church simply wasn’t keeping any before the 1840s).
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project and can offer FTDNA testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).