Hi RajahNZ
Jessie was born 16 Feb 1856 in Stirling.
She's with her mother, brother James and her grandmother (Jean Stirling) in 1861, living at Cowan Street, Stirling. Still living in the same street in 1871 (different house number shown) with mother, grandmother (Jane Martin - same grandmother, but this is the grandmother's maiden name) and all three siblings as listed below.
When did she leave Scotland? There's a possibility for her on the 1881 census but the year of birth is out by four years and she's in lodgings at Coatbridge, occupation a milliner. Might not be her though if she's left for NZ already. If she hasn't, she's certainly not with her family because in 1881 they're living Springburn, Glasgow.
Alexander's occupation is down as ship carpenter on Jessie's birth extract. He appears on later censuses with Jessie's mother despite not being listed on the two censuses above.
Siblings:
James 1 Jan 1859
John 29 May 1861
Mary Jane 9 Aug 1865
Marriage for Alexander King and Janet Stirling - 5 Jun 1852 Lecropt, Perth (also registered in Dumbarton).
Janet Stirling's parent's marriage - Jane Martin and James Stirling - 25 Oct 1823 Lecropt, Perth, Scotland
Jessie's father died in 1903 and her mother in 1904.
If you PM me your email address I'll send you Jessie's birth extract and the death extracts for her parents.