Your hyperbilirubinemia is near the cutoff point of being considered "Conjugated". This means that the likely cause can be determined by looking at the reasons for Direct Bilirubin elevations. It also means that your liver cells (hepatocytes) are doing they job, but the bilirubin/bile transport process is not quite right for some reason. I assume that your liver enzymes levels (ALT, AST) are normal since you haven't listed them. If they are significantly elevated as well, then several forms of hepatitis could also be the cause.
Typical extrahepatic causes are impeded bile flow, from your liver to your small bowel. This can be simply due to stones, bile duct infection/inflamation, etc. There are a number of causes that your doctor will need to sort through in order to diagnose you. Intrahepatic causes are usually due to compressive hepatobiliary issues, from an enlarged liver. This could be toxic hepatitis (drugs), viral hepatitis, etc.
Keep working with your doctor and hopefully the diagnosis will be found quickly. You will likely have some imaging done and maybe additional blood work. But your levels are high enough that you need to get this done relatively soon. Too much bile backing up into the liver can cause significant damage.