Excellent exploded view from AppliancePartsPros to help identify the likely parts.
The Jenn-Air stovetop burner "module" can be unplugged, and once that was done some relatively simple disassembly of the burner module with a phillips head screwdriver was all that was needed to get access to the parts. The spark igniter needs some angling to get it into place, but once done it screws back down.
Be careful with the gas supply lines, on this model they're simple rigid pipes, but they do have to be re-aligned or adjusted when reassembling the module.
A suggestion might be, based on the age of the stove, to include some new screws for the spark igniter. They're small and it looks like over 10 years they've worn down a bit.
I spent $75 with shipping to fix a part that would have been almost $300 to replace. Well worth the effort.