First, we need to calculate the amount of tax Jesse will need to pay. To do this, we multiply the food cost by the sales tax rate: $105.79 * 0.09 = $9.53
Now we need to calculate the pre-tax amount Jesse will use to calculate the tip. To do this, we subtract the tax amount from the food cost: $105.79 - $9.53 = $96.26
Now we need to calculate the tip amount. To do this, we multiply the pre-tax amount by the tip rate: $96.26 * 0.20 = $19.25
Now we add the tax amount and the tip amount to the food cost to find the total cost: $105.79 + $9.53 + $19.25 = $134.57. Answer: \boxed{134.57}.