Your log says it can't find the Java 6 jdk, and it tries to look in a jre folder. Are you sure you installed the JDK (different than the JRE)?
[LOG] JAVA_HOME is set? true
[LOG] JAVA_HOME = null
[LOG] JAVA_HOME Exists? false
[LOG] JAVA_HOME points to a valid JDK? false
[ERR] No JDK found. Attempting to install or ask for the JDK.
[LOG] We're on Windows, searching for the JDK in default locations.
[LOG] These locations are: [Ljava.lang.String;@21f9dd
[LOG] Looking at this location... C:\Program Files (x86)\Java\
[LOG] Looking at this location... C:\Program Files\Java\
[LOG] Location is a directory... C:\Program Files\Java\
[LOG] Looking at this file... C:\Program Files\Java\jre6
[LOG] Looking at this location... C:\Java\
[LOG] Nope, there is no JDK. Ask the user now.
[LOG] JAVA_HOME is now: null
[ERR] Still no JDK is installed. Can't proceed. Either user canceled JDK installation or something is wrong on the user's end.
[LOG] Aborting Android build.