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Use double quotes instead of single quotes to expand the shell variables:
sed -n "/$FromTime/,/$ToTime/p" $logFile > new_log.log

The command you tried can't work for at least the following reasons (there may be others):

when comparing dates, you need to compare year, month, and day (in that order); not day, month, and...

This is a duplicate of another thread you started: Error while extracting data from log file...

Welcome to the forums. Have you tried something?

Based on your code:
last -R|awk -v v1=`date +'%h'` -v v2=`date +'%d'` '$5==v1 && $6==v2'