Previous hacker cut out main support for joist. What is the correct way to splice/lap into the good construction and continue with a double 2X10 header. I'm thinking of triming back 1 ft on the good section to tie this together and put a steel plate over the lap joint on both sides.

Thanks

You need to either place columns under the beam where it is spliced or replace it. Without knowing the loads on the beam it is not wise to make any other suggestions for repair to this critical support member.

Splices in main girders are almost impossible to do. Your choices are to add posts and footings as Home Crae club suggested, or to replace the girder.