Unable to initialize W5100 using Arduino Pro Mini
One of the Ethernet examples shows you should test the result of the begin function call, like this:

// start the Ethernet connection:
  if (Ethernet.begin(mac) == 0) {
    Serial.println("Failed to configure Ethernet using DHCP");
    // try to congifure using IP address instead of DHCP:
    Ethernet.begin(mac, ip);
  }


Try displaying the return value from begin to see if that initialization succeeded.