To fix the problem you need to get inside the printer (involving quite a few screws, a lot of fiddling about and quite a bit of inky mess!). It's officially a 'technician only' job, which means that it would probably cost more than a new printer.
I can't see why you don't want to use compatible cartridges. As long as you use a reputable brand they're just as good as the 'real thing' (particularly if you're not printing photos). But,since you insist, Kodak printers are often the cheapest to run. The ESP C110 has very good reviews:
http://www.reevoo.com/p/kodak-esp-c110
It costs between £50 and £60, depending upon where you purchase it from.
Chris