You can report the loss

• at any identity card authority, usually the local citizens' office, or
• at any police station.

You must also apply for a new ID card if you are over 16 years old and do not have a valid passport document, i.e. a passport or temporary passport. This is possible at the citizens' office at your main place of residence.
If you do not apply for your ID card at your main place of residence, you need an important reason. There is also a surcharge of EUR 13.00 on top of the fee. If you contact your chosen Citizens' Registration Office in advance, you can find out whether and to what extent the Citizens' Registration Office will accept your reason.
In the event that you find your ID card again, you must report this personally to your ID card office. Only when you have officially reported your ID card as "found" will a corresponding entry be deleted by the police.
If you have not applied for a new ID card, you can continue to use your ID card in Germany without restriction until it expires.