Velocity induced by a straight vortex filament
You need to think in 3D. A vortex will induce a flow speed everywhere.

That is the point  lies in the plane containing the flow induced by the vortex at 

Why would you think that vorticity induces a speed only in one plane? You start from the wrong presupposition.
Now consider the rings around the vortex filament in your first drawing. All around those rings a flow speed is induced. Its magnitude drops with the inverse of the radius of one particular ring but is constant along the circumference of that ring. Actually, that ring is only a section through an infinitesimal tube, just as infinitesimal as that vortex filament.

Irrotational vortex. © Silver Spoon, picture source.
Also, the symbol for vortex strength is a capital Greek letter Gamma ($\Gamma$), not $T$.