that will depend very heavily on the time of day, but basically you are crossing through one of the world's busier cities. It'll be a lot quicker at 9pm than 9am.
The AA routefinder gives the distance as 6.0 miles and the driving time as 23 minutes.
The journey from Waterloo International to Liverpool Street is 2.4 miles (14 minutes) and usually costs �12 to �14. So you might expect to pay somewhere between �20 and �25 for your journey. (�1 per minute usually provides a reasonably accurate guide to London taxi fares).