Your plant is a Dendrobium. It's hard to tell what type from you photo. I think you did the right thing by giving it more sun. Just be careful you don't over do it. The damage on the leaf tips could be small areas that were exposed to sunlight longer than other areas of the plant. The canes aren't shriveled and the roots look good so it's probably receiving adequate moisture. Too much moisture is not good and will rapidly cause the roots to rot. The new cane looks healthy and straight. The other one probably grew sideways when it was searching for more sunlight. It would not be wise to remove it because it would rob your plant of energy.
I think your Dendrobium looks healthy but we need to determine the type so you can care for it properly. There are some varieties that are persistent (keep their leaves all year), bloom from the top or near the top, prefer warmer temperatures, and can be watered sparingly but regulary during winters. There are other types that are deciduous (loose their leaves in winter), bloom from leaf nodes along the length of the cane, prefer intermediate to cool temperatures, and require little or no water during winter dormancy.
I would not be concerned about the lack of blooms for now. Different plants bloom at different times of the year. It may not be the right season for your's to be blooming. I live in Central Florida and grow Phalaenopsis type Dendrobiums and most of them bloom in summer or fall. Have you ever seen the plant in bloom? You could describe to us where the bloom occured on the plant if you don't have a photo. What growing zone do you live in (where do you live)? The hairy cane thing is an important tell tale sign too. A close up shot would be helpful.