Would one be absolved of responsibility by using a government microscope for personal work, then repenting for that and donating its value to serve the sick?

It is not permissible for an employee to use work tools and equipment for personal work unless there is explicit or customary permission. What you did, replacing a work device with another without permission, is not permissible. However, it is hoped that you will be excused due to your ignorance of its impermissibility and that your remorse counts as repentance. To absolve yourself of this transgression, it is sufficient to pay the value of the device or the missed benefits resulting from its use, along with regret and seeking forgiveness. The money should be deposited into the hospital's treasury, or if that is not possible, a device for treating patients should be purchased with it. You are not required to buy a new device because you did not damage it, but if you do, you will be rewarded.