You should not mix server-side implementation with client-side implementations. The standard ASP.NET DropDownList control is a server control and doesn't support search functionality out of the box. Here's what you can do:
If you want to implement server-side approach then you can try out:
AJAX Control Toolkit ComboBox[
^]
If you want to implement client-side approach, then use jQuery Autocomlete:
AutoComplete DropDownList Alternatives In ASP.NET[
^]. You can also use the jQuery Select2 as you mentioned.
There are plenty of examples on the web that demonstrate how to implement it. Just doa quick search at google about "DrodownList Autocomplete in ASP.NET" to get more examples.