It's going to have to be elbow grease off when you heat the coils.
First, gently pull them out of your stove top and, using baking soda as a cleanser, scrub any burnt stuff or food that's left on the coils. Use soapy water.
Note: be careful never to immerse any electrical connection to get wet.
I use baking power to clean any gunked up motorbike steel parts that I service. Baking powder acts as a really kewl cleanser, so there's no reason why it can't sort your stove's electric coils.
First, make a paste with baking soda and water and let it sit for 30 minutes, then scrub. That's going to be your elbow grease! Afterwards, be sure to rinse them thoroughly under the tap.
I'd say that unless those coils are scratched or, over time, worn, then there's no reason why they can't bull up nice and lovely and clean. After that, perhaps I should employ you as a garage hand.