I've done my fair share of burning things, but this always seems to work for me - first add water and vinegar (50/50) to cover the burned food. Leave as long as possible, at least over night, but better if you can manage for 24 hours.
After soaking, rinse with hot water and scrub as much residue off as possible with a plastic 'scrubbie'. Finally, add a cup of table sugar and turn it into icing sugar using speed 10 for around a minute. This method leaves my bowl shiny and bright.
Hope it helps you.