Try using a binary collation first, which will mean that the complex Unicode rules are replaced by a simple byte comparison.
SELECT COUNT(*)
FROM MyTable
WHERE MyColumn COLLATE Latin1_General_BIN2 LIKE '%some string%'
Also, have a look at chapter titled 'Build your own index' in SQL Server MVP Deep Dives written by Erland Sommarskog
The basic idea is that you introduce a restriction to the user and require the string to be at least three contiguous characters long. Next, you extract all three letter sequences from the MyColumn field and store these fragments in a table together with the MyTable.id they belong to. When looking for a string, you split it into three letter fragments as well, and look up which record id they belong to. This way you find the matching strings a lot quicker. This is the strategy in a nutshell.
The book describes implementation details and ways to optimise this further.