You want

sqrt{(dx/dt)^2 + (dy/dt)^2}

So

(dx/dt)^2 = [-e^{-t}cos(t) - e^{-t}sin(t)]^2

= [e^{-t}cos(t) + e^{-t}sin(t)]^2

= e^{-2t}cos^2(t) + 2e^{-2t}sin(t)cos(t) + e^{-2t}sin^2(t)

= [e^{-2t}cos^2(t) + e^{-2t}sin^2(t)] + 2e^{-2t}sin(t)cos(t)

= e^{-2t}[cos^2(t) + sin^2(t)] + 2e^{-2t}sin(t)cos(t)

= e^{-2t} + 2e^{-2t}sin(t)cos(t)

Similarly:

(dy/dt)^2 = [-e^{-t}sin(t) + e^{-t}cos(t)]^2

= e^{-2t} - 2e^{-2t}sin(t)cos(t)

So

sqrt{(dx/dt)^2 + (dy/dt)^2}

= sqrt{e^{-2t} + 2e^{-2t}sin(t)cos(t) + e^{-2t} - 2e^{-2t}sin(t)cos(t)}

= sqrt{2e^{-2t}}

as you said you needed.

-Dan