I’m adding an outlet above our fireplace and the “easiest” way is fishing a 14/2 with ground NM down 24” from the attic. Very close to where I want to go fishing is an outlet run in the attic. What’s the best way to tap into this run? I’ve drawn up my thoughts on a whiteboard and attached them here. The wire I’m tapping into is pulled super tight so the only way I found was to add 3 jumpers into the system. Please let me know your thoughts if this is the way to go. The circuit I’m adding to is lightly loaded with only 7 outlets currently on the 15A breaker. Please note the two wires marked existing in the drawing were once one continuous wire I’m “breaking” to add to. The “squares” inside the box in the drawing are wire nuts
You got the right idea but all those wires and splices in a 4"box will be tough. If there's no slack, the existing cable will only be 2" into the box on each side. Think about using two 4" boxes and connect them together with a 2" nipple or piece of EMT. That will give you plenty of room to work with. You'll be running and splicing 14'2 with ground. Your drawing shows 14'3, I think you mean 14/2 with ground.
Each lead coming into a box must have enough cable sheath to get onto the clamp and 1/4" past the clamp (more is better for marking purposes). And then the wires inside the box must continue beyond the sheath for 6" minimum.
You can't do it the way you propose, unless the box is 14" long lol. When you want to mid-splice like that, your better bet is 2 boxes set about 16" apart.
My own preference is an EMT "dogbone", with 2 steel boxes, 2 EMT fittings and an appropriate length of EMT conduit, then THHN white and black individual wires inside the conduit. (black can be re-marked to serve as red etc.)
You may want to consider an NEC 334.40(B) compliant splice kit, like a NSi NMT-2 splice. https://images.homedepot-static.com/catalog/pdfImages/49/4927386c-f90e-4fc6-b8ae-6d113682d82a.pdf