There's an article in this month's Garedners' World - view it at http://www.gardenersworld.com/plants/features/flowers/how-to-grow-agapanthus/5592.html
One 'theory' is that Ags that aren't performing in the ground should be dug up and potted, if not doing well in pots put them in the ground. As you say it's "in the garden", I'm assuming that it's in the ground, and as it's not doing much it may be worth digging up later in the year (work carefully away from the crown to find the extent of the rootball) and potting up (splitting into smaller plants if necessary)

How much sun does it get? They like at least half a day, more if possible, so if yours is in a shady area, that would cause this trouble. Do you feed it at all, and if so, how often and what with (they usually flower without any feed at all....)

It is in a very sunny south facing border which loses the sun in the later afternoon. It is in clay soil,..have fed it with seaweed once, but someone told me they thrive on neglect, so have not fed it again. Do also have a small white Agapanthus in a pot,..but that has only managed two flowers. Have threatened large plant with removal if it does not start to produce more flower heads, as it is taking up a large area of border. Does that count as neglect??

Well, they need good drainage so won't like being in clay. And feeding once won't do much. To get flowers you need to use a high potash (potassium) feed - seaweed is ok for that but on its own isn't a balanced fertiliser (in the absence of anything else a tomato feed is as useful as any for flowers) and feed every couple of weeks throughout spring and summer.

Well, they do quite well in clay, I grow them here successfully in a clay soil, but they do not like to dry out. Their preferred conditions are fertile soil that doesn't dry out completely, but isn't waterlogged. If you didn't improve the soil before planting by adding composted materials, that's something that would help, and that's true for the majority of plants growing on heavy clay soil.

I suggest you apply a mulch of well rotted animal manure from the garden centre, a good couple of inches deep - do it in autumn, when the plant is starting to die down. No need to dig it in, though you can if you want to, otherwise just leave it surrounding the plant, it'll get taken into the ground on its own. Next year, feed with a balanced liquid fertiliser (Miracle Gro or similar) weekly or fortnightly during the growing season, but stop feeding once the flowerheads appear (assuming they do).

You don't have to feed with a liquid fertiliser, it sounds like some soil improvement and/or irrigation during dry periods is really what's required, but if you want to encourage it, its worth a try.