Currently, I have that for a X-Band High Gain Antenna functioning at 8Ghz, diameter of 3 meters, and a power transmission of 13.1dBW(20Watts) at a distance of about 8.5AU has a Gain of 52.44dBi, free path loss of 292.59dBi, and a total received power from transmitter of -152.13dBw. I am very new to the realm of telecommunications, so I'm not sure if these are acceptable values for the distance. I am trying to find specifications that would allow for a decent transmission rate from Enceladus to DSN.
Doing a scaling process from the performance of the telecommunications ("telecom") system of NASA's Cassini spacecraft at Saturn, I estimate the system you describe could support a data rate of a bit under 7 kbps, if the signal is received at a NASA DSN station's 34-m (diameter) antenna. If a 70-m antenna were used instead, that rate would jump to nearly 29 kbps — but getting time on the 70's is not easy! Using a different receiving network can affect the indicated data rate because the receivers' performance figures differ.
Note that there are system characteristics you haven't mentioned that can cause a few dB of difference in overall performance. For instance, your computed on-axis gain looks a bit high (my scaling calculator indicates ~48 dBi). One such characteristic is the antenna efficiency (which essentially considers losses due to subreflector occultation, subreflector support struts, imperfect conductivity of the primary and secondary reflectors, and a bunch of others) that decreases the gain from the ideal. Another is the way the primary reflector is "illuminated": how does the intensity of the radio energy incident onto the primary reflector vary as you move from the center to the edge of the reflector? Depending on that pattern you get different beam shapes and different gains.
There's a huge amount of information about telecom systems and their design at JPL's Descanso web site. It's not for the faint at heart, but you seem to have sufficient background to start on their material.