The last job I got called out to was because the customer had a dribble coming out of the basin tap and the bath tap.
The first thing you should do is check you have mains pressure in the kitchen, because this is fed from the mains. If you have then you know that your problem is internal and has previously been suggested .. it is a blockage in the tank or pipework leaving it.
You have already said you have a suspect lid on the tank.
Turn the mains off ..
Get a torch and have a look inside the tank in the loft ..
If it is manky .. most are I'm afraid .. you will need to drain it down and clean it out.
Remove you mixer above the bath and drape it down into the bath, remove the shower head to aid drainage..put the plug into the bath.
Turn the cold tap on and see what arrives in the bottom of the bath, you will be very surprised at how much rubbish comes out.
When the tank has drained down then you will need to remove any sediment that has gathered.
It is then a case of filling the tank and flushing it out until you clear.
Another cause of a slow water flow could be the ball valve having a blockage within it or the washer having perished.
Having found decomposed rats, pigeons and mice in tanks, it wont surprise me what you find up there.
A word of warning... it is always advisable to screw a plank across 2 joists so that you have a good platform to work on as most lofts are a dangerous and unfamiliar place to work.
...... set a spotlight up if you have one.