You can get LED vanity bulbs on Amazon. They’re on a string and attach to the mirror with a command strip, they’re also dimmable. Something like this

https://www.amazon.com/Pretmess-Hollywood-Adjustable-Brightness-LED/dp/B07RZCT9DH/ref=mp_s_a_1_7_sspa?crid=1D03FGTI3OBG2&keywords=vanity+led+bulbs&qid=1669597592&sprefix=vanity+led+bulbs%2Caps%2C103&sr=8-7-spons&psc=1&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUE5WFZGWklZVE1KWlAmZW5jcnlwdGVkSWQ9QTAwODE0NTczMDFFREpQRzhPVEs5JmVuY3J5cHRlZEFkSWQ9QTA5NDg5ODVCUjVKNEkxQTdEWjEmd2lkZ2V0TmFtZT1zcF9waG9uZV9zZWFyY2hfbXRmJmFjdGlvbj1jbGlja1JlZGlyZWN0JmRvTm90TG9nQ2xpY2s9dHJ1ZQ==