Probably you want:
SELECT ID FROM Tasks WHERE task LIKE '! @ # $ \% ^ & * ( ) ''%' ESCAPE '\'
"in the database, the data in the field is Today's Task"
Check again - I just created a SQLite DB, with a table Tasks and three entries:
ID(int) Task(nvarchar(50)
1 Hello
2 Today's Task
3 XX Today's Task XX
NULL NULL
Then I hacked some C#:
string strCon = @"data source=D:\Temp\TestingSQLite.sldb";
SQLiteConnection con = new SQLiteConnection(strCon);
SQLiteCommand cmd = new SQLiteCommand("SELECT * FROM Tasks WHERE task LIKE '%Today''s Task%'", con);
con.Open();
DataTable dt = new DataTable();
SQLiteDataAdapter da = new SQLiteDataAdapter(cmd);
da.Fill(dt);
And got two rows: 2, and 3.
Check your data!