Lamps not working after blown bulbs
Occasionally when a lamp blows the elements short and create a large amount of heat the center conductor may not be touching or the switch may have failed.
The fact 2 lamps blew in a short period may be due to power spikes in your area but unless you have other household items not working verify there is a connection from the cord to the center then the other side of the cord to the shell of the lamp holder. If the center is not working on either side of the cord and turning the switch on still provides no continuity the switch is probably bad and will need to be replaced.
A cheap 10$ ohm meter is all you need to check the cord/switch combination and you can even check the bulb to see if it is good by measuring from the center to outside.