Finding one power wire to two lite switches. ugh!
I think I know what's going on here, the solution is simple if the situation is as I understand it. I'll go over it again as confirmation.

The butchered wire should be the unswitched power from source that continues through the two switches, one for each light in turn, then on to the lights, then return via neutral. If you're in North America and the usual color codes were followed, this wire should be black. (Neutrals are white)  The original configuration had a short length of insulation stripped from the end and attached to a switch (closet?). A few inches away, the same wire had a length of insulation stripped, but with insulation left in between the stripped portions. The copper wire itself remained continuous, the non-end bare portion looped partly around the binding screw of the other switch (bathroom?). See sketch below "Original". Only the pertinent wiring is shown, all other wiring omitted for clarity.

You then taped up the inner stripped portion and reinstalled the wire end to one switch, leaving the other without power. Correct?

If so, this is a somewhat common bad practice, mainly because the unbroken partial loop cannot be properly bound to the terminal. The fix is easy. You need a wire nut sized for 3 conductors of whatever wire gauge is used (usually #12 or #14 AWG) and two short lengths of the same gauge black insulated solid copper wire.

Remove your taped patch and cut the wire so that the entire wire is completely insulated except for the final 5/8"-3/4" which remains bare. (See sketch "Cut") Attach this and the two short pieces (ends stripped in similar fashion) with the wire nut.

The other ends of the short pieces are attached, one per switch, to where the original wire was attached. (See sketch "New") There is a particular way to make binding screw connections which you may not see in the existing work. Do NOT make a simple U bend and hook it around the binding screw, you do not get adequate surface contact this way. (Sketch "NO!")

Instead, pre-bend the wire end into a nearly complete circle, so it is configured much like an eye bolt eye. (Sketch "Yes") Re-open the end gap just enough to slip the binding screw through. The loop must go clockwise around the screw so tightening the screw closes the end gap. Before tightening the screw, pre-close the gap as best you can with needle nose pliers. Firmly tighten the binding screw and it will draw the gap the rest of the way closed.

You can see electrically you have the exact same situation, but now you have used good quality methods to achieve the connections.