EF core Using a generic factory to register a DB context

I am out to sea with this.

I received this answer which was corrected here by Nkosi.. however whilst it indicated how to rewrite the factory I am not up to the task of implementing it.. I need some gaps filled in.

As I understand it, this is what I need to do..

define the interface:

public interface IContextFactory<TContext> where TContext : DbContext 
{
    TContext Create(string connectionString);
}

Create the factory:

public class ContextFactory<TContext> : IContextFactory<TContext>
    where TContext : DbContext
{
    public TContext Create(DbContextOptions<TContext> options)
    {
        return (TContext)Activator.CreateInstance(typeof(TContext), options);
    }
}

register the factory as a singleton in startup.

This is what I did and I dont know if this is correct?

services.AddSingleton(typeof(IContextFactory<>), typeof(ContextFactory<>));

Next how the heck do I use this factory?

In the first answer, it was suggested I use this:

public class EntityBaseRepository<T> : IEntityBaseRepository<T> where T : class, IEntityBase, new()
{
    private JobsLedgerAPIContext _context;

    public string ConnectionString { get; set; }

    public EntityBaseRepository(IContextFactory<JobsLedgerAPIContext> factory)
    {
        _context = factory.CreateDbContext(ConnectionString);
    }

    public virtual IQueryable<T> GetAll()
    {
        return _context.Set<T>().AsQueryable();
    }

    public virtual int Count()
    {
        return _context.Set<T>().Count();
    }
}

Nkosi identified that you cannot have the constraint "new()" and a parameterized constructor. He rewrote it. How do I change the above code to reflect the fact the factory now does not have a parameterized constructor but still take in the connection string given Nkosi's factory?

ClientRepository inherits the above.

public class ClientRepository : EntityBaseRepository<Client>, IClientRepository
{
    private new JobsLedgerAPIContext _context;

    public ClientRepository(JobsLedgerAPIContext context) : base(context)
    {
        _context = context;
    }

Does this constructor need to be changed.

And finally how do I Inject the ClientRepository now that I have to supply a connection string?