The closest I can find for Elizabeth is Elizabeth Ann Halford, widow, born Shardlow c1815, who is in the county asylum in 1881 & 1891.
She was buried 10 Apr 1901 at Welford Road, aged 82, putting her birth at c1819, which more closely matches the 1841 & 1851 census dates for Elizabeth

Finding a marriage would be a start, failing which you will need to obtain a birth certificate for one of the children to establish Elizabeth's maiden name.