You need to launch faster
10.37 km/s is just about the escape velocity from Venus, meaning that the object thrown away from Venus at this speed at surface would lose most if not all of its speed while leaving the planet's gravity well. Thus the answer for this speed is about infinity, or rather in the basspark of 10^21 seconds as you would then tear off the significant part of Venus' mass thus lowering the escape velocity, making more impulse count. Yet, if you would launch mass faster, the times would be able to get calculated.
Let's say your Lofstrom loop is actually encompassing the entirety of Venus' equator, being connected to the surface in at least two locations opposite each other, and the launch point of that loop is located above Venus' atmosphere. This configuration could potentially launch projectiles at higher speeds, limited by tensile strength of the loop's belt. Here on Earth the loop is believed to be able to operate at sub-escape velocity only, yet let's say your civilization has made discoveries allowing the loop to operate at no less than 12.0 km/s, sustaining launches of one-ton masses per second. This device will allow those masses to escape Venus at speeds of sqrt(12.0^2-10.36^2) ~= 6.055 km/s, essentially applying an integral force of 10^3*6055*10 = 6.055e7 N at level length of 6051.8 km plus the loop's height (arbitrary, but can't be too high in order to allow the loop to ever start - let's say 100 km), or 6.152e6 m. This amasses to 7.725e14 N*m, at least a factor of 100 better than what the proposed solar reflectors could do (that answer had a 0.5x mistake in radius, read 6051 km as diameter instead), thus the expected time to accelerate Venus would be correspondingly smaller - 1.435e18 seconds, or 4.55e10 years. Still too slow. Note that during all these years Venus would lose 1.435e21 kg of mass, or 0.0295% its total mass, making escape velocity slightly lower over the process and speeding up Venus' rotation slightly more effective.
Note that increasing final speed would only lessen the gravity-related momentum losses, without any cardinal increases in the acceleration speed (at most effective change of momentum would be equal to mass multiplied by starting speed), thus increasing linear speed of projectiles would only give linear effect on time required to speed Venus up, at most to double (12/6.055) effectiveness, so having that mass driver to launch stuff at 100 km/s would shorten the required time by a factor of 2*100/12~=16.66, thus even while this idea looks quite more attractive, it would still not allow that civ to terraform Venus before it'll get consumed by the Sun.
(Perhaps next time calculate the idea of having ion engines at space towers? These don't require as much mass down the drain as the exhaust speed is quite high, and you can make quite a lot of those around Venus' equator.)