$60K credit card debt (sick child had to quit job)
Should I look at pulling this from my IRA, 401K, or a loan against the mortgage?
Income is too low to keep up payments and I am hoping to prevent falling behind on payments.
First of all I sympathize with your situation. I, too, would do anything for my children, and I can't imaging having to sacrifice as much as you have. That said, you have a big hole now and need to find a way to dig your way out.
Income is almost enough to pay the bills but nothing left for discretionary items...like food.
Food is NOT discretionary (restaurants are, but not actual food). That needs to come first. Then comes housing, then utilities. By utilities I mean electric and water, not cable, internet, and telephone. Those can be cut temporarily until you get out of the hole. You can't afford to be "comfortable" right now.
Look at local organizations for assistance. Many churches have food banks that can help with other needs (like clothes and utilities) as well. You might have to swallow your pride for a while, but you might find someone who can help you out.
Borrowing against the house or retirement funds does not solve the problem, it just moves the debt. It may mean a lower interest rate, but you need to jack up the principal payments somehow.
Cashing in retirements would solve the short term problem, but you're going to pay the price in the long term since you won't be able to retire until much later in life. Another drastic option is selling your house and finding a cheap apartment. That would be my last resort before bankruptcy.
You can survive by moving debt around, but you'll be paying payments the rest of your life. Hopefully you can find more work as your child's health improves and can rebuild your income, which is the only way you're going to get rid of the debt and thrive instead of survive.
If you're currently running a "lean" budget (and from your comment, it seems you are), and it would be balanced by a lower interest 401(k) loan or second mortgage, then yes, it seems a reasonable path to take.
I'd look for a home equity loan (aka second mortgage).