Two months is still really not much time for noticeable improvements, especially if you didn't run before. Your body adapts slower than you'd like. But it's not just about time - it's about how you're training. If you run at the same pace the same distance every time, your body gets used to it fast and stops improving - that's why there's no progress and your breathing isn't getting better.
Try mixing up your workouts: one day easy, easy running (when you can still talk in sentences), the next day with speed work or intervals (alternating fast and slow), the third day aerobic again but a bit longer than usual. This approach forces your body to adapt in different ways and builds both endurance and speed. Your breathing improves when your body learns to use oxygen more efficiently - and that happens specifically with varied workouts.
Also pay attention to recovery: how much you're sleeping, how you're eating, maybe you're burnt out from overtraining or stress. And don't get hung up on your 5k time - it'll come. First focus on how it feels: your breathing should get easier, running should feel more comfortable. The numbers will catch up in about a month or so, once your base gets stronger.