At the hospital where I work, different assistants use different techniques, but if I am ever asked to bathe a patient, this is what I do:
1. Gather all your things. Soap, Shampoo, Deo, 2 basins--1 for soapy water, 1 for clean warm water for rinsing, lotion, towels, & washcloths. I don't care for the "Ready-baths" that you can buy. They are like baby wipes with some type of cleanser in them. You just micro them, wash, and dry. No rinsing. Many people like these, but I don't personally. I only use these in the peri areas (aka private parts).
2. Take mine and Juliana's advice and cover your MIL with a sheet or light blanket. Only expose that part of the body you are washing while keeping the rest of her covered (it's chilly to them).
3. Start at her face and work your way down.
4. Keep a dry towel under the body part you are washing. This keeps the mess at bay. Dry each part as you bathe and rinse it to lessen shivering.
5. Wear gloves each time you bathe her. Put them on beforehand and not just for the private areas.
6. I wash hair last and use the shampoo cap as mentioned above. Just micro them for a few seconds. If you don't have these, you could put baby shampoo on a washcloth and massage the hair, then rinse. Baby shampoo is less foamy than regular adult shampoo. I only use a small amount. Always put a rolled up towel under the head/behind the neck to catch spills.
If possible, I would recommend putting a hand-held shower attachment on your shower head and buying a shower chair. If she is able to sit in a chair, she could sit in the bathtub while you helped her bathe. This technique is so much more effective. You can find these at Goodwill stores for way less than new ones. Just bleach it beforehand!
Sometimes, we are asked and called to do things that seem impossible and embarrassing to us--especially in the medical field. However, I find that if you can spawn a pleasant conversation throughout the procedure, it makes you and them feel a little more at ease. If you are a person of faith, you may ask her if you can pray with her before the bath in specific. This sometimes helps my patients to relax.
Also, if you really want to go all out, you can buy some bath tools, like hand-held massagers, frozen eye masks, cucumber slices for the eyes, foot scrub, play relaxation music from Pandora or youtube or other soothing music while you bathe her. Do a complete "spa", if you will. Get different loofahs and that type of thing.
I hope this helps you and you are able to become a little more comfortable in an uncomfortable situation. Just think of this from her perspective...she probably feels just as uncomfortable as you do!