Scott,
You find Kajsa and her daughter in Skällebyn in Åmåls landsförsamling (Åmål rural parish) and the husförhörslängd volume AI:3 1804-1810 on page 40 (GID 659.4.57000). On that page, her birth place is said to be Knöttkärr or Knöttkjärr as it is spelled in the index. She comes from No 109 in 1806, but I have not figured it out what this means. It does not correspond to a plot number in the city, nor to a page in the husförhörslängd; it might be a number in the flyttlängd which unfortunately seems to be destroyed for the years before 1823.
The information that Kajsa is born in Knöttkjärr indicates that she might be the daughter of the soldier, later corporal, Nils Knös who you find with his family in Åmåls landsförsamling husförhörslängd (which I will abbreviate to Ålfs hfl) AI:1 1784-1790 in Västra Knyttkärr on page 8 (GID 659.2.14300). Here you find Kajsa born in 1780. The hfl is missing for the years 1791-1797, but Nils is still living at Västra Knyttkärr and has now risen to the rank of corporal (Ålfs hfl AI:2 1798-1805; GID 659.3.34300). Here Kajsa has already left the family and I have not looked for her; I'm afraid that you have to start at page 1 to find her.
Nils and family moves to Björkil when he retires somewhere around 1799 and the family is noted on page 41 (GID 659.3.36800). They don't stay long in Björkil and moves the same year to Källeby (page 62; GID 659.3.38900) and here Kajsa reappears, moving in from (V?) Åmål in 1799. Her birth year seems first to have been noted as 1784 and then crossed out and changed to 1774. The birth year of Nils, 1755, might make Kajsa's birth year plausible, but her mother's, 1760, indicates it must be wrong.
The family moves a lot in 1799 and they are found on page 136 (GID 659.3.46300) and Risa. Kajsa, still born in 1774, moves to the city in 1801 and you find her in Åmåls stadsförsamling husförhörslängd AI:5 1799-1804 on page 364 (GID 1964.48.97700). Here you see that her birth year has changed to 1782, but the reference to the page she was moving from and the year as well as her birth place, shows that it is the same Kajsa. She the moves to No 83 which poses the corresponding problem as with your Kajsa, i e where she moves from in 1806.
I do find it probable, even though the birth years vary, that your Kajsa Nilsdotter is the daughter of Nils Knös. Maybe if you go through every page of the husförhörslängd of both the city parish as well as the rural parish (AI:5-6 and AI:2-3 resp.) you can establish the whereabouts of Kajsa from 1801-1806.
Her correct birth year, though, is the one noted in Ålfs AI:1, i e 1780. You find her birth record here: 1964.1.5700, the place says Knöttkierrs stomb.
Well, I hope that you will be able to tie together the information by a more closer examination of the husförhörslängd, and if you do it would be interesting to know!
/Lars