Oldgoat I have had a bit of a look on FreeBMD and then some general internet searching. My results are here (a lot of guesswork, but maybe you can follow some of it up and see whether there is anything in my gueses)
FreeBMD is just England/Wales, not Scotland, so if she was Scots searches will have to be at Scotland's People.
1) Searched FreeBMD index for births of a Jenny Whittle between 1850-1890. NONE
2) Thought maybe she was married, so searched for a marriage 1870-1890 between a Jenny X and a Y Whittle - NONE
3) Jenny is Scots diminutive for Janet, so searched for marriage Janet X to Y Whittle - YES - Janet Linton married Thomas Whittle, Easington (ref 10a 697) 2nd quarter of 1890
4) So searched for birth of Janet Linton - YES - first quarter of 1867, Houghton, Co Durham (ref 10a 424)
Linton is a Scottish placename (East Linton and West Linton not far from Edinburgh) Maybe she had Scottish parents/ancestors and was visiting to see them as well as doing the Grand Tour?
5) Internet search for Eagle Hall in County Durham - YES Eagle Hall Farm, Sunderland Road, Hawthorn, SDeaham, SR7 8RU
None of this confirms that it is the same person - but it may give you somewhere to start from.