How do I link an image in my plugin so it displays on WordPress?

This line is your problem:

$new_site_name = $site_name . '<sup>echo '<img src="' . plugins_url( 'images/wordpress.png' ,"logo.jpg"  ) . '" > '</sup>';

Here you’ve muddled up the quotes and put PHP code inside a string expecting it to still work.

e.g.

echo ' this is inside a quote:' inside a quote ' . ';

It doesn’t work because the quote in your string closes the starting quote, ending the string. To fix this you need to either:

• use a double quote
• escape your characters

e.g.

echo 'this is inside a quote \' inside a quote \' .';

As a side note, make sure that your code editor supports syntax highlighting with a good colour scheme, it will make these kinds of issues more obvious.