It sounds like what has happened is that part of the plant survived and part didn't. On the part that has those long bloomless canes it's the root stock coming up. Most root stocks are once a season bloomers on old wood so they won't bloom until next season, if the canes survive this winter, and the bloom will not be your original rose but probably a dark red single rose called Dr. Huey.

Since you do still have part of the plant that is your original rose, and I assume you want to keep that, you need to take that root stock growth off right away. If left to grow it will eventually kill the other plant and take over.

In order to get rid of that root stock permanently you have to dig down to where those canes are attached to the root ball and RIP them off. Do not just cut them with pruners because they'll simply grow back again. Tearing them off should damage and destroy the growth cells so they can't come back up. You have to do damage in that spot.

It sounds to me more like winter damage. The canes are healthy enough to leaf out, but not healthy enough to produce healthy leaves and flowers. The remedy is pretty much the same though. Prune it out. However, the description doesn't match either or the two common rootstocks.

Pictures would help us come to a definitely conclusion.

It the non-flowering canes are rootstock, they would have grown from underground this season and would still be growing vigorously without setting flower buds. You would probably see differences in the leaves apart from size and color--different shape or number of leaflets, different degree of gloss. In this case do as Seil said.

If they are winter-damaged canes of the (original) scion variety, they would be old canes from last year that are not supporting good growth this season. Leaves could be paler, smaller and bunchier than normal. The canes would produce short stems that go blind (stop growing without setting a flower bud). In this case, just prune the canes to the ground as Mad Gallica said.