There are reasons why this wouldn't have worked at all. I'll explain at the end, but first the numbers you ask for.
Liquid mass in Orbiter plumbing:
- Common feedline (between external tank disconnect valve and engine prevalve): 4000 lbm LO2, 250 lbm LH2
- Engine feedline (between engine prevalve and SSME): 298 lbm/line LO2, 22 lbm/line LH2
Using the engine flow rates from an earlier question of 925 lbm/s LO2 and 154 lbm/s LH2 at 104% and the same proportional assumptions, I get 3 engines running at 109% depleting the LH2 in 0.65 seconds and the LO2 in 1.68 seconds. (Bringing up more reasons why this wouldn't have worked)
Single engine running at 67% assuming it uses all the common manifold and one engine line gives LH2 depletion in 2.74 seconds and LO2 depletion in 7.21 seconds.
Why it wouldn't work:
- Closing the ET disconnect valves on a running engine(s) was considered catastrophic due to the water hammer that would result. Special pneumatic locks were added to hold the valves open after the Challenger failure.
- The propellant trapped in the feedlines would instantly lose pressure and the SSME turbopumps would cavitate.
- Propellant depletion shutdowns were considered catastrophic due to turbopump overspeeding, LO2-rich shutdowns doubly so due to "burning and severe erosion of engine components"
- Assuming #1 and #2 didn't blow up the shuttle, the onboard computers would have commanded all running SSMEs to shut down because the propellant depletion sensors (LH2 in the ET, LO2 in the common feedline) would have gone dry. There was no way to override this shutdown command.
References
- Challenger disaster: how full was the external tank at the time of destruction?
- https://science.ksc.nasa.gov/shuttle/technology/sts-newsref/et.html
- SMS mass properties page (personal notes)
