Originally Posted by silken
Welcome! I agree with vjo. Since it is done blooming, you might also want to re-pot it if the media is old looking. I can't tell from the photo. These Dens. like a mix that drains well and dries out fairly fast. They really like to be pot bound as well. There are lots of sites on line as to the culture of these but basically they like a cool dry winter which is what initiates them to bloom again. Also no plant food thru winter. Since yours just finished, it may be a bit out of sync as they manipulate bloom time in the nurseries to whenever they want. When the new growth is more mature, you can likely get it to bloom this way. They also like quite bright light but not direct sun that could burn them. I wouldn't remove the new growth as most orchids need 3 or more old pseudo bulbs as support to the new growth for good blooming. Just let it grow into a multi growth plant.
For Den. nobile type hybrid, winter is not a good time to repot since they will start to enter their resting season.
I would pull all the dead flowers off. Put it in a sunny location indoors. In the UK, a south face window right next to the glass would be perfect. In nature, they experience a dry winter monsoon that is sunny and cool. Let it have a dry rest. Water may be once a month or so. If you see shriveling in the cane water it a bit more. With this treatment by late winter or early spring you might get another flowering.
Growth season starts in spring, you can gradually start to water the plant a bit more, and start to feed more.
By the look of your picture, there is no need to cut any canes or bulbs.
We grow these outdoors here in Southern California year round. Our winter temps drop to 5 c occasionally but usually stay above 10 c at nights. I put mine in a sunny balcony and forget about it from late November to Valentines day. I mean no water, no fertilizing and they seem to grow just fine.