Arduino Ethernet Shield uses the W5100 Ethernet controller. The Arduino Ethernet library only supports W5100 so you need to install an Ethernet library that supports W5200. Here are 3 options:
Seeed Studio EthernetV2_0: GitHub - Seeed-Studio/Ethernet_Shield_W5200: Seeed Studio Ethernet Shield V2.0 Library It will require a small modification of the Arduino Ethernet code to work with the Seeed library(change #include <Ethernet.h> to #include <EthernetV2_0.h>).
Modified version of the Arduino Ethernet library that supports W5100, W5200 and W5500 with automatic detection: GitHub - per1234/EthernetMod at W5x00 that will not require any modification of the Arduino Ethernet code to work with the W5200.
WIZnet Ethernet Library: GitHub - Wiznet/WIZ_Ethernet_Library: WIZnet Ethernet Library You must modify src/utility/w5100.h to specify you are using W5200.