Welcome to Fluther!
I am also not a Doctor. I have a Bachelor’s Degree in Speech and Hearing Science, and a Master’s in Audiology. I work in an Ear, Nose and Throat practice, and I also used to sing. My advice is making the assumption that you have a normally functioning, healthy immune system, and no chronic illnesses. If your immunity is compromised or you are under a physician’s care for any reason, you should definitely contact a doctor.
As noted by @JLeslie, aspirin is acidic, so you will feel irritated for some time. If you cleared what solid material was remaining you should gradually feel better over the course of a few days, and should be all the way better within two weeks. If you think there were still bits of aspirin lodged laryngotracheal area, you may feel a bit worse over the next day or two before you start to feel better. If you choose to wait it out, but don’t feel like it’s betting better by Monday, then definitely see an ENT who can look down your throat, trachea and esophagus with a laryngoscope.
In the meantime, keep your body well hydrated and nourished, and avoid acidic or spicy foods, to reduce the risk of further irritation. Keep the area moist by avoiding dry air and breathing in steam.
Try to get rest your throat and vocal cords by avoiding coughing and throat clearing. They happen almost reflexively, but you can mentally override the urge by reminding yourself that the irritation is from inflammation and not an object in your throat. Coughing and throat clearing create friction in the throat which causes irritation and inflammation; then as the tissue tries to heal, it can get itchy... so we cough and clear our throat – it’s a vicious cycle. Slow, controlled breathing and drinking hot herbal teas with honey can help soothe it.
If you can take a different anti-inflammatory NSAID, such as ibuprofen (Advil/Motrin) or naproxen sodium (Aleve/Anaprox) they might help reduce some of the inflammation and irritation.
I hope you recover quickly, and we appreciate updates on Fluther.