Answer: I think the plant you have is botanically named Epiphyllum oxypetalum. This is an interesting plant in that it does not really need soil to live. The most common reasons for lack of bloom with this plant are lack of light and too much nitrogen fertilizer and lack of a "winter" rest period.
In a bright south facing room I would expect the light to be enough (bright indirect is adequate).
You could reduce the fertilizer and see if that helps; if your plant is a good green color it is probably oversupplied with nitrogen. Nitrogen excess can cause a plant to grow but at the expense of blooms. You would not need to fertilize at all in winter or summer as the plant grows primarily in spring and fall. Try something like a water soluble 5-10-10 with minors at maybe a quarter of the label rate when it is in active growth. Less nitrogen should also slow down the vegetative growth and help keep it smaller.
As far as cutting it back, this plant is naturally large and rangy so it will be an ongoing problem. You might try cutting it back (if it blooms, prune right after the bloom period) to a reasonable size and let it regrow. Keep in mind the flower buds will appear along the flatter side branches or shoots so do not remove all of those.
To help the plant set buds, it needs a cool winter rest period with short days. The temperature should be a bit cooler than a normal home would be, say between 45 and 50 degrees during the winter months. Since it is cooler, it will need a bit less water as well.
There should be no artificial light in the room with this plant during the winter --it needs uninterrupted darkness at night. This will allow the plant to experience short days as it would in nature during the winter. This primes it to bloom once the days begin to lengthen and spring arrives. I am not sure how practical this will be since it is in your dining room, but it may be the key. In other words, a barely heated, unused south facing room without lighting is a good place to keep it during the winter.
I hope this helps!
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