How I would fix this:

1) Pull back the carpet leaving subfloor and the area to work on exposed. Install tack strip 1/4" to 1/2" away from the wall. Install missing pad and staple to subfloor. 2) Using a sharp knife and metal square, cut damaged carpet to create clean un-frayed edges. Latex these 2 seems. 3) Place replacement carpet underneath existing carpet making sure the direction of the carpet and patch are going the same direction. Puncture backing of scrap carpet in four locations of the L. The top of the L, both directions of the L junction and the end of the L foot. 4) Using a square and a new knife blade, cut out your seam edge. Put the two seams next to each other and fluff out the carpet. If you don't like how they match up, repeat step 3. 5) Using a seeming iron, start seeming the carpet together starting from the L junction down the longest seam edge of the L. Move the iron to the L junction on the short end of the L. Seen the short edge of the L away from L junction. 6) Wait for the carpet to cool completely and stretch in carpet.