Natalie,

My recollection of the play is that the Nurse was the sort of woman who would cheerfully have carried the message to Juliet herself. But let that pass...

he stands 10m from the house and lobs a small pebble at her bedroom window (3.5m high).
Romeo throws the pebble from a height of 1m with a speed of 11.5m/s at an angle of 60 degrees
to the horizontal. I have already found that it take 1.74 seconds to reach the window and
that it does in fact hit juliet's window

A reasonable target: not as deep as a well, nor as wide as a church door, but 'twill serve. Now, how to find $v_T?$

To find that it hits the window (without symmetry tricks that don't apply here) you must presumably have found the horizontal and vertical components of the pebble's velocity at launch.

The horizontal velocity is unchanged [we ignore air resistance] and the vertical velocity is found using the formula $v = v_0 + at$ (remember a is negative) for which you know all the RHS terms at impact.

Now use Pythagoras' formula to find the length of the vector with those components. Any remaining error (if not arithmetic) is then probably due to you and the setter using different values for $g.$

NOT ("Swear not by the moon, inconstant orb") varying tidal forces.

Good hunting!
RD