diff --git a/jwt/jwks_client.py b/jwt/jwks_client.py
index 120eb256..e83f84c8 100644
--- a/jwt/jwks_client.py
+++ b/jwt/jwks_client.py
@@ -87,7 +87,7 @@ def get_signing_keys(self, refresh: bool = False) -> List[PyJWK]:
         signing_keys = [
             jwk_set_key
             for jwk_set_key in jwk_set.keys
-            if jwk_set_key.public_key_use in ["sig", None] and jwk_set_key.key_id
+            if jwk_set_key.public_key_use in ["sig", None]
         ]
 
         if not signing_keys:
@@ -95,7 +95,7 @@ def get_signing_keys(self, refresh: bool = False) -> List[PyJWK]:
 
         return signing_keys
 
-    def get_signing_key(self, kid: str) -> PyJWK:
+    def get_signing_key(self, kid: Optional[str]) -> PyJWK:
         signing_keys = self.get_signing_keys()
         signing_key = self.match_kid(signing_keys, kid)
 
@@ -117,7 +117,12 @@ def get_signing_key_from_jwt(self, token: str | bytes) -> PyJWK:
         return self.get_signing_key(header.get("kid"))
 
     @staticmethod
-    def match_kid(signing_keys: List[PyJWK], kid: str) -> Optional[PyJWK]:
+    def match_kid(signing_keys: List[PyJWK], kid: Optional[str]) -> Optional[PyJWK]:
+        if kid is None:
+            if len(signing_keys) == 1:
+                return signing_keys[0]
+            else:
+                return None
         signing_key = None
 
         for key in signing_keys: