The evidence from trials of acupuncture for urgency and frequency of urination are relatively positive, although the best evidence comes from studies in children and people who are recovering from a stroke. Balance is a different matter. There are a great many reasons why someone's balance may have been lost, ranging from neurological problems and minor infarcts (small strokes) in the brain to problems with the inner ear and occasionally something as trivial sounding as crystalline particles affecting the movement of the little hair-like sensors in the tubes of the inner ear. Research would have to be targeted on a specific cause, and it would be unwise to pass comment drawing on available research without knowing what the likely cause was.
From a Chinese medicine perspective there are some very well-defined syndromes which take the descriptions which patients give of their balance and urinary problems and make sense of them within the overall functioning of the body. In some cases there are distinct treatment protocols which have been developed over centuries to try to address these problems, and the trained practitioner will look for conformation from the signs which they observe in the pulse and tongue, as well as other symptoms which to a western medic may appear to be unrelated. Even if there is no clear cut pattern, Chinese medicine was initially premised on the belief that symptoms were only the expression of a complex set of inter-related imbalances in the system, and the practitioner's task was to use his or her skill to interpret the evidence they gathered and set about correcting imbalances in the simple belief that a system in balance does not generate symptoms.
Clearly the best advice is to visit a BAcC member local to you to discuss with them whether acupuncture could be beneficial for your specific case. We hope that they will give you an indicative assessment which will help you to choose what your best treatment options may be.