What does it say about adding to knob and tube.
"300.16 Raceway or Cable to Open or Concealed Wiring.
(A) Box, Conduit Body, or Fitting. A box, conduit body, or terminal fitting having a separately bushed hole for each conductor shall be used wherever a change is made from conduit, electrical metallic tubing, electrical nonmetallic tubing, nonmetallic-sheathed cable, Type AC cable, Type MC cable, or mineral-insulated, metal-sheathed cable and surface raceway wiring to open wiring or to concealed knob-and-tube wiring. A fitting used for this purpose shall contain no taps or splices and shall not be used at luminaire outlets. A conduit body used for this purpose shall contain no taps or splices, unless it complies with 314.16(C)(2)."
Install a box into which to terminate the new cable, leaving enough wire stripped to extend from the holes in the other side of the box to an insulator on each knob and tube conductor. Mount that box about 1 foot away from a set of knob insulators. Using a wood block or similar fulcrum and a pair of diagonal wire cutters
pull the nail that secures the two halves of that insulator to the framing out enough that you can fit the wire from the new cable into the unused slot on the insulator. Place loom over the wire from were it starts in the box through the holes by which each conductor leaves the box all the way to the insulators that you will add the new wire to. Since woven fabric loom is no longer available You will need to use flexible plastic tubing for the loom. The split type of plastic loom is not acceptable because it is too easy to disturb and accidentally remove from the wire. The flexible plastic tubing provides physical protection to the wires. It is not insulation and does not need to be listed by a testing laboratory. Use a size of tubing that is just large enough to get the individual wires through and not something an inch or more in interior diameter. Secure both the original wire and the new wire in the insulator by gently tapping the nail through the middle of the insulator until it is snug. If you strike the nail too hard the insulator will shatter. Scrape a small length of insulation off of the knob and tube wires ~4 inches past the insulator. Cut the two wires from the box so that they are just long enough to go through the insulator slots and reach the bare spot on the original knob and tube wire. Strip the insulation off the 2 new wires about 1 inch from each end. Connect the two new wires to the bare spots on the knob and tube wiring with split bolts sized for the conductors. The split bolts should have a size range that has the size of the wires being used which is as close the the largest wire that the split bolt is marked for as practicable. Use a side cutting plier
to hold the bolt head still and tighten the nut with a nut driver.
The joint must be snug but not so tight as to cut into the wires. Tape the split bolt with 3 layers of plastic electrical tape. Find a way to run your new Equipment Grounding (Bonding) Conductor (EGC) to the nearest junction box on a circuit that has an EGC. Run that new EGC into the box where the added on length of nonmetallic sheathed cable comes off of the knob and tube wiring.