find matching text and replace next line
One way: Sample file
$ cat file
Cygwin
Unix
Linux
Solaris
AIX
Using sed, replacing the next line after the pattern 'Unix' with 'hi':
$ sed '/Unix/{n;s/.*/hi/}' file
Cygwin
Unix
hi
Solaris
AIX
For your specific question:
$ sed '/<key>ConnectionString<\/key>/{n;s/<string>.*<\/string>/<string>NEW STRING<\/string>/}' your_file
<key>ConnectionString</key>
<string>NEW STRING</string>
This might work for you (GNU sed):
sed '/<key>ConnectionString<\/key>/!b;n;c<string>changed_value</string>' file
!b
negates the previous address (regexp) and breaks out of any processing, ending the sed commands, n
prints the current line and then reads the next into the pattern space, c
changes the current line to the string following the command.
It works. Additionaly is interested to mention that if you write,
sed '/<key>ConnectionString<\/key>/!b;n;n;c<string>changed_value</string>' file
Note the two n's, it replaces after two lines and so forth.