and your mother simply cannot hold a pen in one hand (or two hands) and just sign the form on a dotted line???????
Some people manage to write holding a pen with their mouth or toes...
Sorry to hear about her conditions, but it just seems a bit weird that she cannot sign a form because of arthritis, a wheelchair and a breast lump.
As for the public charge - don't worry about it, you are signing an affidavit of support for the rest of her life, promising to support her at 125% of poverty line and to repay federal government benefits she may get (there are restrictions, read the affidavit).