Try installing the java6 plugin.
If that doesn't work you may need to update your default java version or manually create a symlink to the java plugin.
sudo apt-get install sun-java6-plugin
If the java link is pointing to the wrong version then update it with the following. Make sure to replace the paths below with one of the paths that were displayed from the commands above.
update-alternatives --list java
update-alternatives --display java
If the java path is correct then try creating a symlink to the java plugin. I don't currently have Epiphany installed so I can't tell you the exact plugin path but it would look something like the following.
sudo update-alternatives --install "/usr/bin/java" "java" "/usr/lib/jvm/jre1.6.0_14/bin/java" 1
sudo update-alternatives --set java /usr/lib/jvm/jre1.6.0_14/bin/java
sudo update-alternatives --install "/usr/bin/javaws" "javaws" "/usr/lib/jvm/jre1.6.0_14/bin/javaws" 1
sudo update-alternatives --set javaws /usr/lib/jvm/jre1.6.0_14/bin/javaws
And finally, test your java versions.
sudo ln -s /usr/lib/jvm/jre1.6.0_14/lib/amd64/libnpjp2.so /usr/lib/firefox-addons/plugins/libnpjp2.so
Java.com also offers a simple java applet test. Just click the 'Do I have java?' link on the homepage.