Will I be sinful for not visiting my ill, bedridden mother more than once every two weeks, knowing that I am busy with my family and work, and that there is a long distance between our homes?

The status of a mother is great, and honoring her is an obligation, and earning her pleasure is an act of closeness (to Allah). For, "The pleasure of the Lord is in the pleasure of the parent, and the wrath of the Lord is in the wrath of the parent." Since you have done what you can to maintain ties with her, you have fulfilled your duty. However, you must be patient with her reproaches and beware of mistreating her, lest you fall into disobedience (to parents). You should strive to earn her pleasure, such as giving her gifts and offering financial assistance, so that you may thereby attain Allah's pleasure and be happy in this world and the Hereafter.