You've got the right parents for Anna - her mother is listed as "
svärmor", mother-in-law, below and said to be the widow after Per Andersson. Her birthyear also matches the age given in the birth notice.
Jan is said to be born in Djörkshyttan -- and he can be tracked back to his parents, and paternal grandparents, in AI:12b, p 101 (mp)
[1]. The place-name is spelled "Görkshyttan", but pronounced the same -- "
Jœrkshyttan" (in my home-made phonetic script ...)
Tracking people in those records is made easy by the fact that they apparently married young, and stayed in place -- and they were marked in the books with their relationships: "måg" (son-in-law), "svåger"/"svägerska" (brother/sister-in-law), etc.
--
[1] Hedemora (W) AI:12b (1800-1811) Bild 106 / Sida 101 (AID:
v130365.b106.s101, NAD:
SE/ULA/10388)
[2] Hedemora (W) CI:7 (1797-1810) Bild 144 (AID:
v130441.b144)