Maybe you meant for the extension $L/K$ to be algebraic, in which case it is true that any extension of the trivial valuation on $K$ to $L$ is trivial.  This clearly reduces to 
the case of a finite extension, and then -- since a trivially valued field is complete -- 
this follows from the uniqueness of the extended valuation in a finite extension of a complete field.  Maybe you view this as part of the sledgehammer, but it's not really the heavy part: see e.g. p. 16 of 

http://math.uga.edu/~pete/8410Chapter2.pdf

for the proof.  (These notes then spend several more pages establishing the existence part of the result.)