On the pins that haven’t soldered well, is there solder stuck to those pins already, or is there just bare flux? If there is solder there on one of the pins, try heating the pin for a few seconds, to see if the existing solder melts. You can check (for example) with the tip of a hobby knife or wire, to see if the solder is molten. If there is no solder wet to the pins, but just flux, you might consider scraping at the flux on the pins with a hobby knife to expose some metal. Make sure to hold your soldering iron against both the plated hole on the circuit board and against the pin of the component. Make sure the plated rings on the circuit board around the component pins are still in place. Excessive heating or pressure can damage the board and cause these to delaminate.
Once you finish soldering, I’d advise first testing with a standard test circuit, such as the LED blinker. If you can’t get that to work, double-check your soldering. If things still aren’t working, please read the documentation for the kit, which is extensive, and available here: http://wiki.evilmadscientist.com/555
Using that documentation, you should be able to trace through the entire circuit to figure out exactly what is and is not working, assuming that you have a voltage source and a multimeter.