One easy thing to do: dump the old one, getting the dump as MyISAM.
On the new site, create a TEMPORARY database and import the MyISAM tables to *THAT* database. [If the dump file from the old DB specifies the database name, just use an editor to change it in the dump file.]
Now just construct a query that will copy the data from the temp database table(s) into the InnoDB tables in the main database.
Might be tedious if there are lots of tables and lots of fields, but probably lees work than any other way.
When done, you can just drop the temp database.
An optimist sees the glass as half full.
A pessimist sees the glass as half empty.
A realist drinks it no matter how much there is.