SELECT substring(value, charindex('(', value) + 1, charindex(')', value) - charindex('(', value) - 1)
FROM string_split('North Carolina (01) >> State (03) >> State Funded (09) >> AMI (21)', '>')
WHERE value LIKE '% State Funded %'
Although I predict that there are strings that do fit this pattern. This kind of stuff is not very fun to do directly in SQL Server, unless you want to write a CLR function to wrap the RegEx class.