Are there companies that offer assistance with patent filing and are willing to cover the associated fees in exchange for partial ownership? Can you recommend some such companies?
You may want to consider filing a provisional patent application which is a pretty low hurdle to gain at least temporary protection for your invention. With a provisional filed, you can approach the standard sources of funding such as venture capital firms, angel investors or established companies in the same field. Intellectual Ventures is a company solely dedicated to acquiring and then licensing patents and patentable ideas. It has its critics and some consider it a patent troll. There may be similar companies, but I'm not aware of them.
As for specific recommendations, I'm pretty sure that would be off topic here. I, at least, would not feel comfortable providing one.