If you have an ANSI encoded string, using utf8_encode() is the wrong function to deal with this. You need to properly convert it from ANSI to UTF-8 first. That will certainly reduce the number of Unicode escape sequences like \u0082 from the json output, but technically these sequences are valid for json, you must not fear them.
Converting ANSI to UTF-8 with PHP
json_encode works with UTF-8 encoded strings only. If you need to create valid json successfully from an ANSI encoded string, you need to re-encode/convert it to UTF-8 first. Then json_encodewill just work as documented.
To convert an encoding from ANSI (more correctly I assume you have a Windows-1252 encoded string, which is popular but wrongly referred to as ANSI) to UTF-8 you can make use of the mb_convert_encoding() function:
$str=mb_convert_encoding($str,"UTF-8","Windows-1252");
Another function in PHP that can convert the encoding / charset of a string is called iconv based onlibiconv. You can use it as well:
$str=iconv("CP1252","UTF-8",$str);
Note on utf8_encode()
utf8_encode() does only work for Latin-1, not for ANSI. So you will destroy part of your characters inside that string when you run it through that function.
For a more fine-grained control of what json_encode() returns, see the list of predifined constants(PHP version dependent, incl. PHP 5.4, some constants remain undocumented and are available in the source code only so far).
Changing the encoding of an array/iteratively (PDO comment)
As you wrote in a comment that you have problems to apply the function onto an array, here is some code example. It's always needed to first change the encoding before using json_encode. That's just a standard array operation, for the simpler case of pdo::fetch() a foreach iteration:
while($row=$q->fetch(PDO::FETCH_ASSOC)){foreach($rowas&$value){$value=mb_convert_encoding($value,"UTF-8","Windows-1252");}unset($value);# safety: remove reference$items[]=array_map('utf8_encode',$row);}