Hi there, sorry you haven't got a reply sooner,
Shite して is the -te form of suruする, "to do".
Shimasuします is the polite present tense of suruする, "to do".
Gozaimasu ございますis from gozairuございる, which is a very polite verb meaning "to be".
Eg. "Ohayou gozaimasu", literally "HONORIFIC-Early IS(polite)" ("hayou" is from "hayai", meaning "early")
Eg 2. "Arigatou gozaimasu", literally "Thank you-IS(polite)".
Nasaiなさい is the imperative form of nasaruなさる, a very polite verb meaning "to do". It is used as a auxiliary after the -masu base of verbs.
In "Machi nasai!" まちなさい！The -masu base of "matsu", "to wait", is まちますーー＞まち。
nasaiなさい is then added as an imperative auxiliary adjective, adding that assertive tone,"Wait!". It is quite polite. You could also say "matte!", and "matte kudasai!" to mean the same thing.
Watakushi わたくしis simply a formal version of watashiわたし.
If there are any errors, someone please correct me.
Hope this helps!