Chronic Pyelonephritis or Persistent Urinary Tract Infection (UTI): The patient's symptoms of frequent urination and back pain, which persisted despite antibiotic treatment, suggest a possible chronic infection of the kidney (pyelonephritis) or a persistent UTI that was not fully eradicated by the initial antibiotic course.

Kidney Stones: Frequent urination and back pain can be symptoms of kidney stones, especially if the stones are causing obstruction or irritation in the urinary tract.
Interstitial Cystitis: A condition causing bladder pressure, bladder pain, and sometimes pelvic pain, which could explain the persistent symptoms despite antibiotic treatment.
Overactive Bladder: Characterized by a sudden urge to urinate, which could be a contributing factor to the frequent urination symptom.

Pyonephrosis: A serious condition where the kidney collects pus due to an infection, often due to an obstructed urinary tract. It's crucial to diagnose and treat promptly to avoid severe complications.
Sepsis from a Urinary Source: If the urinary tract infection has spread to the bloodstream, it could lead to sepsis, a life-threatening condition requiring immediate medical attention.
Tuberculosis of the Urinary Tract: Although less common, TB can infect the urinary tract and cause chronic symptoms similar to those described.

Bladder Cancer: Could present with symptoms of frequent urination and back pain, although it's less likely in the absence of other symptoms like hematuria.
Genitourinary Fistula: An abnormal connection between the urinary tract and the reproductive system, which could cause continuous urine leakage and other urinary symptoms.
Endometriosis Involving the Urinary Tract: Endometrial tissue can rarely implant in the urinary tract, causing cyclic symptoms that might include frequent urination and pain.