Yes, I think she's from the mine agent Dixon family, youngest child of William Dixon:
William Dixon / Mary (Mally) Thompson, m 1 Dec 1782 Crosthwaite
- William 1751/2 - 30 Mar 1837
- can't find b/c
-> children:
Margaret Dixon - c 21 Dec 1783 Borrowdale
John Dixon - c 4 Aug 1788 Borrowdale
- d Mar1866 aged 77 (Cockermouth 10b 331)
William Dixon - c 1 Nov 1791 Borrowdale
Dinah Dixon - c 14 Sep 1794 Borrowdale
- m William Birkett, 8 Aug 1825 Crosthwaite
Joseph Dixon - c 24 Apr 1797 Borrowdale; d 18 Feb 1807
Daniel Dixon - c 11 Dec 1799 Borrowdale
Jane Dixon - c 5 Jul 1801 Borrowdale
Jane married Thomas Birkett on 13 Nov 1827, Crosthwaite
Thomas was one of the many children of Joseph Birkett (1755-1836), farmer at Seathwaite, Borrowdale:
Joseph Birkett / Dinah Stockdale, m 24 May 1790 Crosthwaite
- Joseph b 27 Mar 1755 Crosthwaite, d 8 Aug 1836 Seathwaite - see will
- Dinah (1769-1850)
Mary Birkitt - c 5 Nov 1791 Borrowdale, d 8 Dec 1791
Sarah Birkitt - b 28 Oct, c 5 Nov 1792 Borrowdale
- m - Charles Stuart 9 Apr 1817 Millom; d Threlkeld Jan 1870
John Birkett - c 4 Jun 1794 Nether Wasdale (Joseph Birkett / Dianah)
- m Hannah Bainbridge 10 Jul 1819 Muncaster
William Birkett - c 22 May 1796 Nether Wasdale
- see below
Nancy Birkett - b ? Mar 1798 (not in IGI)
- m - Jenkinson
Dinah Birkett - c 11 May 1800 Irton
- m - Benjamin
Betty Birkett - c 27 Jun 1802 Thwaites
- m - Thompson
Jos Birkett - c 23 Sep 1804 Thwaites
Thos. Birket - c 21 Sep 1806 Thwaites
- m 13 Nov 1827 Jane Dixon (qv, b 1801), in Seathwaite c1841-71
Mary BIRKETT, b ca. 1809
- m Thomas Bennett 1 Apr 1828 Crosthwaite.
Jane BIRKETT, b 11 Dec 1811, Broughton in Furness;
- m William BURROWS 20 Jun 1833 Crosthwaite.
Hannah BIRKETT, b 1817, Broughton in Furness;
- m William SIMPSON 25 Apr 1844 Crosthwaite.
There is more info on both families, and the transcript of Joseph's will of 1836, which mentions 7 daughters and 3 sons including Thomas as beneficiaries, that I can point you to; and the Dixon family are mentioned in Tyler's book, Honister Slate.
Best wishes, Denis