How do I solve this network of diodes?
In principle all combinations have to be checked, however often a few simplification can be made by inspecting the circuit.

The diode with the voltage U6 across it will always be on. There is no possibility to bring the anode below 0.7V.

The cathode of the diode with U7 is off because there is a resistor between the cathode and ground. If it was on there would be a voltage drop across the resistor reducing the voltage across the diode. This voltage is 0.7V minus the voltage drop across the resistor. However, below 0.7V it would turn off.

The diode with U3 across it is on. It has 8V at the anode, the remaining 7.3V split equally across the resistors. We can verify that the diode with U7 is still off.

The currents are a result of the applied voltages, they are not needed for anything.