You're welcome. I did find what I'm fairly certain is Anna's birth record. From the HH exams we know that her birth date was January 20 1859 and that she was born in Slagesnäs. And in the HH exam for Kyrkhult 1865 - 1875 she is living in Slagesnäs with John Persson (b. Feb 22 1818), Bengta Olsdotter (b. Nov 10 1833), brother Ola (Mar 1 1863), and sisters Elsa (Sept 17 1866) and Bengta (April 19 1871). Which fits in nicely with her surname being Johnsdotter - but John Persson and Bengta Olsdotter weren't married until Dec 31 1864. Looking back through the birth records Ola's parents are listed as John Persson and Bengta Olsdotter, but the only Anna born on January 20 1859 had parents listed as (unmarried) Anders Persson and Bengta Olsdotter, age 25. I did find Bengta Olsdotter and daughter Anna living in Slagesnäs in 1859 (both birth dates match) and followed them through to her marriage to John Persson in 1864.
Perhaps John Persson adopted her? Perhaps if you posted to the Swedish board someone there could explain why she had the surname Johnsdotter rather than Andersdotter....? And just as a by the way - you'll find the pre 1865 the Kyrkhult records included in the Jämshög records.
Best of luck,