I'm sorry that your question wasn't answered in a timely manner. A loss of balance is a serious matter in parakeets. It's important to note that once a parakeet acts ill they're quite ill and in need of the attention of an avian-oriented vet (please see here: www.aav.org). This is a protective mechanism because sick birds are attacked by other birds in the wild. Emily's symptom of loss of balance can indicate head injury, brain involvement in a systemic disease process, or general weakness but loss of balance isn't pathognomonic (specifically indicative) of any one disorder.
An avian-oriented vet will first treat symptomatically and supportively by providing supplemental fluids and electrolytes by needle and tube feeding a "recovery" food. Blood tests and cultures of Emily's choana - the slit between her oral cavity and nose - and cloaca (vent) may be taken.
Until Emily can be attended to, please heat up her environment to 85F by means of a 100W bulb shined into her partially covered cage (not at night when she needs to rest) or by taping a heating pad set on its lowest setting to the sides of her cage. Remove her perches and put her food and water on the bottom of the cage along with her. Add a water soluble avian vitamin such as Oasis brand to her water at half of the recommended dose so as not to make her water distasteful. Add a calcium supplement such as Calcivet or Calciboost to her water. These supplements are available in pet/feed stores. Avoid over the counter antibiotics designed to be placed in her water. They won't be effective if only because an ill bird won't drink enough to medicate itself properly.
Nutritional imbalances are a common cause of illness in our pet birds. What has Emily's diet consisted of, please? Seeds should compose less than 20% of her diet. Ideally, a balanced pelleted diet such as can be found here: www.harrisonsbirdfoods.com or here: www.lafeber.com/pet-birds should be fed as well as hard boiled egg yolk, pancakes and cornbread, the tops of fresh greens, dairy products such as yogurt and cheese, fresh fruits such as apples, pears, melon, kiwi, and berries, vegetables such as broccoli, carrots, beets, asparagus, cabbage, sweet potato, and squash, and even tiny pieces of meat.
Please respond with further questions or concerns if you wish.