I was so down and unmotivated, nonetheless, that even thinking about getting up the chair seemed like a tedious and burdensome task.
I'm having my doubts because at first I used however, and know, well, I'm not very sure anymore.
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nonetheless is most often used as a replacement for however, in spite of or despite that so it doesn't sit well in your sentence.
For nonetheless to be useful in your sentence, the second part really has to be in opposition to the first as in I was so down and unmotivated, nonetheless, I clambered on to the chair with the spriteliness of a chimp.
Or a preceding sentence is in opposition such as I'd just won the lottery but I was so down and unmotivated nonetheless.
so much so would work better in your sentence.
I was so down and unmotivated, so much so, that even thinking about getting up the chair seemed like a tedious and burdensome task.