"Please close the window."
I think the hint for 'close' is showing the hiragana for the intransitive polite verb 'shimarimasu', as in "the window is closed", where the TE form in the answer comes from the active verb? This might help: https://www.japan-guide.com/forum/quereadisplay.html?0+109792
There's only two irregular verbs in Japanese する to do and くる to come. The verb used in this sentence is 締める(しめる) to close. しめて is the te form of しめる. Duo is recommending the wrong verb in the tips. Not that しまりますdoesn't also mean to close, but that it is not the verb that is being used in this particular sentence.