Short answer: The motherboard is bad. I find that common on the L505D, unfortunately.
Long answer: From what I’ve seen with these, the Toshiba L505D is affected by the good old lead-free solder failure issue that Xbox 360s and many computers have. I’ve saved at least one, if not two or more L505Ds following [https://www.eevblog.com/forum/beginners/getting-started-with-bga-reflow-and-reballing/msg472940/#msg472940|this reflow procedure] with great results. But I’d only pursue this if you or someone you know has good experience with soldering. It’s important to note that a preheater, in my experience, helps the results to last much more than otherwise. If you don’t have one, I wouldn’t bother.
Regardless, it is an older laptop, so I’d look into replacing it. If you’re looking for something inexpensive, there are many great refurbished laptops out there that can be had for less than a brand new one, and as far as data transfer goes, all you need is a SATA-to-USB adapter to plug your L505D’s HDD into your replacement laptop. It’s easy to get to the HDD on the L505D.
Regardless of which method you choose, good luck!