问题可行的解决方法是检查API响应，并查询请求的实例状态。可以使用wait_until_running（）方法来等待实例运行，并使用recovery_instance_id属性来检索创建的实例的ID。具体代码示例如下：

import boto3
client = boto3.client('drs')

response = client.start_recovery(
    source_instance_id='source_instance_id',
    target_instance_type='t2.micro',
    target_region='us-east-1'
)

recovery_instance_id = response.get('RecoveryInstance').get('InstanceId')

if recovery_instance_id:
    print('Recovery instance created with ID:', recovery_instance_id)
else:
    print('Error creating recovery instance')

# Check instance state and wait until it is running
waiter = client.get_waiter('instance_running')
waiter.wait(InstanceIds=[recovery_instance_id])