I know that this is an old one, but I had the same dramas.  Since a more recent version of web jobs you can use instances and instance methods, and pass in custom IJobActivator instances. It's surprisingly easy.
It works perfectly with Ninject.  I haven't seen any Ninject examples, so ...
public class MyJobActivator : IJobActivator
{
    protected readonly IKernel _kernel;
    public MyJobActivator(IKernel kernel)
    {
        _kernel = kernel;
    }
    public T CreateInstance<T>()
    {
        return _kernel.Get<T>();
    }
}
public class MyBindings : NinjectModule
{
    public override void Load()
    {     
        Bind(typeof(DbContext)).To(typeof(MyEntities));
    }
}
class Program
{        
    static void Main()
    {
        using (IKernel kernel = new StandardKernel(new MyBindings()))
        {
            var jobHostConfiguration = new JobHostConfiguration
            {
                JobActivator = new MyJobActivator(kernel)
            };
            var host = new JobHost(jobHostConfiguration);
            // The following code will invoke a function called ManualTrigger and 
            // pass in data (value in this case) to the function
            host.Call(typeof(Reminders).GetMethod("ManualTrigger"), new { value = 20 });
        }
    }
}
public class Reminders
{
    private readonly IMyService _myService;
    public Reminders(IMyService myService)
    {
        _myService = myService;
    }
    // This function will be triggered based on the schedule you have set for this WebJob
    // This function will enqueue a message on an Azure Queue called queue
    [NoAutomaticTrigger]
    public async Task ManualTrigger(TextWriter log, int value, TextWriter logger)
    {
        try
        {   
            // process the notification request
            await _myService.FindAndSendReminders();
            await _myService.SaveChangesAsync();
        }
        catch (Exception e)
        {
            logger.WriteLine(e.Message);
            Console.WriteLine(e.Message);
            throw;
        }
    }
}
EDIT: In addition to the above I've seen recently learned that you may not need to use the  host.Call(typeof(Reminders).GetMethod("ManualTrigger"), at least for continuous web jobs.
You simply make your Functions class non static and add a constructor for injection, and then make your processing method non static. This is Illustrated below. 
public class Program
{        
    static void Main()
    {
        using (IKernel kernel = new StandardKernel(new MyBindings()))
        {
            var jobHostConfiguration = new JobHostConfiguration
            {
                JobActivator = new MyJobActivator(kernel)
            };
            var host = new JobHost(jobHostConfiguration);
            // The following code ensures that the WebJob will be running continuously
            host.RunAndBlock();
        }
    }
}
public class Functions
{
    private readonly IMyService _myService;
    public Functions(IMyService myService)
    {
        _myService = myService;
    }        
    public async Task ProcessReminders([QueueTrigger("reminder-requests")] string notificationMessage, TextWriter logger)
    {
        try
        {   
            // process the notification request
            await _myService.FindAndSendReminders();
            await _myService.SaveChangesAsync();
        }
        catch (Exception e)
        {
            logger.WriteLine(e.Message);
            Console.WriteLine(e.Message);
            throw;
        }
    }
}
I adapted my original code from an article I found for Autofac
http://www.jerriepelser.com/blog/dedependency-injection-with-autofac-and-webjobs
See also
Dependency injection using Azure WebJobs SDK?
And for continuous webjobs
http://www.ryansouthgate.com/2016/05/10/azure-webjobs-and-dependency-injection/