Hi Guys, Welcome to the forum,
If you need the joint to be as strong as the original Pc. of L, I would suggest a full pen joint. Any of the prequalified full pen joints out of AWS D1.1 will give you the strength of the original L. A simple butt joint using SMAW(stick), GMAW(solid wire mig), FCAW(fluxcored wire mig) with a 45* bevel on one side and the other side left square with a 0"-1/8" root opening, 0"-1/8" land (B-U4b) should work for your application. You fill up one side and turn it over and backgouge(grinding with the disc turned on edge is acceptable if you don't have access to a carbon-arc gouger) back into your weld from the other side and make sure you have it clean, and fill it back in.
The lap joint, may or may not develope the strength you need, several factors are taken into account when applying a lap joint to achieve full strength. If you go with a lap joint consult an Engineer to help with the calculations.
John Wright