I invested in a friend's business and lost every penny. But I was younger, the friend had less experience, and the business proposal was not worthy of a bank loan. This deal sounds plausible and your personal situation for making the investment sounds excellent. In your shoes, I'd do it for any price we both thought was reasonable, just as you're suggesting. So - what price? Spitballing follows fwiw...
Because the investment is illiquid, it should bring a higher return than the stock market. Similarly, because it has higher risk being a small business, a higher return.
My guess is that each of those factors should be worth 20% to 30% in a decade for the situation you describe. Multiplying, I'd like to see about 1.6x value after a decade compared to a decent stock/bond mix. Using 5% net of inflation as the benchmark, or roughly 1.625x gain by the benchmark, a target value of roughly 2.6x.
Based on your projections, and assuming all numbers in the projection are net of inflation, then: using 10k/year in years 2-10 as typical return plus 5% compounding of proceeds, presumed distributions would have cumulative value of maybe 110k or a bit more. If the value of the business at that point is also 2x of earnings, total return of 130k. Roughly 2.6x your investment, and I used the low end of your projection. So it seems viable to seek a % that would produce the results you describe.
I'd seek the high end of your proposed 10-20% stake if I could, and figure 15% would be healthy. Keep us posted!
PS. My remarks assume your LLC will be structured so that you can have the hands-off experience you hope for. I'm assuming that's some sort of arrangement where you are essentially a limited partner. If your whole stash is at risk for their business decisions, not just your investment, you would need to be at the high end...and I'd still hesitate, or rather seek revision of the proposed ownership terms. I'd consider a lawyer fee to be part of my investment, to make sure the investment has the limited risk profile that I thought. When my friend went de facto bankrupt, I was glad that I had only loaned money, not taken an equity position.