A rule of thumb to start off with for calculating freelance/contract/consulting rates is to assume 1000 hours/year and figure how much you need to charge for what you want to be earning. By that rule of thumb you should be charging at least $120/hour - arguably more because this is your precious free time you're giving up, placing your health and happiness in danger, and you don't really need the freelance gig. If they balk at $120, they were never going to replace your day job and you're better off without them.

If, after rejection, you plan to continue with the client at the same rate, you will need to make sure the negotiation process does not sour the professional relationship moving forward. This means making sure that you're emotionally prepared to be rejected and also negotiate in a manner that does not come off as aggressive or manipulative.

The best way to ask for it is to highlight the clear benefits you have delivered already and to paint a vision of continued benefits moving forward and then ask for your new hourly rate. Ultimately it's as simple as that.

You will need to overcome objections be referring to the "market rate" and positioning yourself as referring to objective external standards.

In lieu of that option, I'd suggest finding a good stopping point (say after a feature is completed) and talk to your client about pursing further work at a different rate. Since they like both you and your work, I wouldn't be nervous about creating animosity—developers are in EXTREMELY HIGH DEMAND and they will be happy to properly compensate you now that you've proven your value.

I will say that it's still much more difficult to get a raise from the same client. Even if you like working with them you should at least see what else is out there for no other reason than leverage during negotiation.