Hi Spacerowa,
The filter wizard supports single supply operation, but does not do any level shifting for you. In other words, it expects the input signal to be centered around the mid-supply voltage. So you'll want to level shift your input signal so that it is around the reference voltage, rather than ground. This discussion has some more information.
You can do this level shift with a passive highpass filter. However note the 2.7k and 1 nF in Stage A of the filter will likely load your highpass filter and give you some unexpected results. You could either buffer with an additional op amp, or use the component sizing slider in the "I want to choose mode" to increase the impedance of the R&C at the input to make it less of an issue.
From you DC conditions description, it looks like the inverting input voltage may have been shifted a bit too much. You stated the DC inverting input voltage is 6.93V. This means we have a difference of 1.08V from the reference (6.93V - 5.85V = 1.08V). The circuit will amplify this difference. It looks like you have a gain of 3.3k/750 + 1 = 5.4V/V. If we multiply the voltage difference by this gain and add to the reference voltage we get: 5.4*1.08 + 5.85 = 11.68. This is right up against your rail. From its datasheet, it looks like the AD8039 needs at least a volt to work correctly, and you are going to want additional room beyond that to accommodate your AC input signal.
Matt
Message edited: Added DC conditions paragraph