Assuming you've already tossed the front taper that you cut off, so you no longer have anything to splice (which can be done by epoxy-splicing the stripped core of one side into the core and coating of the other, btw), you've got a couple of options. You could needle/nail knot on a heavier than normal (for a five weight) leader butt and make a light line bugging line for warm water (or a Clouser-type line). You could add a short loop to the new tip and loop a longish furled or braided leader to make a passable line--not "whisper quiet" but usable for nymphs, streamers, and medium to large dries.
CC