Never heard "yagete" before. Anyway you are basically right. koto is just nominalizing the entire phrase, which is not particularly odd in Japanese, though it more often gets done with "no" rather than "koto". You can see "no" at the end of sentences all the time, particularly in the phrase "n(o) da". Desho just indicates something the speaker thinks is likely true (or likely to be true in the future).
That's at least the context I learned when I asked my Japanese teacher the most polite way versus the most impolite and offensive way to ask "May I speak English?"
[bɹ̠ˤʷɪs.təɫ] Nōn quālibet inīquā cupiditāte illectus hoc agō Yo te pongo en tu lugar... Taisc mach Daró
For the record, yagate "before long" and yagaru (contemptuous verb suffix) are unrelated.
What was the most polite way your teacher told you? "英語でお話しさせて頂けませんでしょうか" / Eigo de ohanashi sasete itadakemasen deshō ka? ("Will I likely not humbly receive permission to speak English (humbly)?")
