Is there any way I can speed up the process?
This transaction now has six confirmations. You should be able to spend the coins from it now. If the coins were sent to a web service then that service should have made them available for you to spend by now. If a web service is withholding them then all you can do is contact its support.
https://blockchain.info/tx/4cfdbb0aacb82ce6ed560dc6634906aadbedd28737923fc3320a122ecdddabef
Who controls the address from the screenshot you posted? Does a web service control it, or is it from a wallet installed on your computer, or mobile?