You will need to rewrite your code to generate a html_table object to pass into the html_writer::table() function call.
The definition of html_table can be found in lib/outputcomponents.php - the comments in there explain exactly what each bit of the object is for, and there are numerous examples in the rest of the Moodle code.
The get_context_instance calls can be fairly safely ignored for the moment, but if you want to fix them (and this will probably need to be done before Moodle 2.7, as I guess they might become errors at that point) - convert them to the appropriate context_XX:instance($id) calls.
e.g.
get_context_instance($courseid, CONTEXT_COURSE) => context_course::instance($courseid);
get_context_instance($cmid, CONTEXT_MODULE) => context_module::instance($cmid);
(and similarly for CONTEXT_COURSECAT, CONTEXT_SYSTEM, CONTEXT_USER, CONTEXT_BLOCK)
It's not completely clear whether you were the original author of this 'mediasite' module you are seeing the error in - if you are not, then you may find it worth contacting the original author to see if they have upgraded it to work with Moodle 2.6. Alternatively, you might be able to find a developer who can do the upgrade for you.