"Mom has a towel, it is so big you can wrap you whole body in it."
I would translate this as "У мамы есть полотенце. Оно такое большое, что в него можно закутать всё своё тело.

Genitive is not ignored here. It shouldn't be here because Genitive is only used to show possession where you in English use "of" for the same purpose, i.e. Genitive is not used with pronouns showing possession such as my, your, his, hers, etc. "Закутать (своё) тело" grammatically is the same type of phrase as "Видеть тело", which is a case for Accusative and pronouns don't change anything here. By the way, in Russian language, such pronouns act like adjectives in respect to nouns, i.e. nouns "influence" their gender, case, singular/plural form, but the pronouns can't make a noun change its case or anything else.

As for the second part of your phrase I would rather translate it as: "Оно такое большое, что в него можно полностью закутаться (or завернуться)" (i.e. It's so big you can wrap yourself entirely in it). I gave a different translation above because of the "тело/тела" issue.