I had to take out two older canes from my DJ because they were not healthy. Now I have a big hole in it:)

If you cut down to the graft, it should spur new basals, but they could take awhile to grow. Since you cut all the canes down, I'm not sure how the rose will respond.

Most climbers can be grown as shrubs. Cut vigorous basals back to maybe 2-3' and make deep deadheading cuts on laterals, leaving maybe 3-5 leaves on a stem. If a cane is arching, cut to an upward-facing bud eye at or inside of the arch. Don't take out whole canes unless they have become unproductive. Once you get enough laterals to make a bushy plant, it will be less eager to produce climbing basals. Traditional pruning diagrams can give useful advice about partially cleaning out the center in spring.

I find that "Don Juan" is notorious for the growing habit you have mentioned and pruning to create more canes to emerge is needed IMHO...Been there and doing that myself...Jeanne

Mandi,
A lot of Don Juans out there are not great roses (although they have great blooms). Getting the basals is time consuming and sometimes frustrating.
All I can suggest is that you have patience. And you might consider planting the rose HomeRun at the base to fill in the emptyness with a rose that holds onto its leaves and makes good pure red single blooms.

ditto on what michaelg said, I kept all of my Gloire de Dijon as potted shrub roses by judicious pruning and keeping long shooters in check.

It's taken about a week-2wks but it has put out about 10 new canes on the outer sides of the three largest central ones.

I'm going to be picky with which ones get to stay, since even if I don't get great show this year, I'm thinking where this plant will be ten years from now. It's a young garden, so time is on my side I hope. (if I don't let undesirable growth muck it up)