Why is "do" used twice in the quotation, "do you do as the Queen has bidden"?

Solution 1:

This is simply a more emphatic (and verbose) form of the second-person imperative: do + you + bare infinitive, which last is in this case another do. This form allows for extra verbal as well as physical emphasis (in performance) on the redirection of Caesar’s speech from Cleopatra to the Nubian slave; this imperative is addressed to the latter.

OED s.v. do v.:

31. trans As periphrastic auxiliary in imperatives.
    a. With an affirmative imperative: adding emphasis or urgency to an entreaty, exhortation, or command. Also formerly (in a command) with a second personal pronoun preceding the main verb. Originally with the main verb in the imperative . . . ; from later Middle English onwards, with the main verb in the undifferentiated base form, usually taken as infinitive.
    . . .
    1684 J. Bunyan Pilgrim's Progress 2nd Pt. ii. 144 Pray do you go along with us, I will be your Conductor.
    . . .
    1871 E. Eggleston Hoosier School-master xvi. 127 Now, Shocky,..do you run ahead.

Solution 2:

It's a misprint. See Project Gutenberg's text:

CAESAR (cutting her short). I speak to the Queen. Be silent. (To Cleopatra) Is this how your servants know their places? Send her away; and you (to the slave) do as the Queen has bidden. (The slave lights the lamps. Meanwhile Cleopatra stands hesitating, afraid of Ftatateeta.) You are the Queen: send her away.