qgis - Deleting 'garbage' nodes after polygon dissolve

I work with QGIS 2.10.1 on Windows 7 (x64). I have dissolved polygons from an attribute value and the polygons geometries being not perfectly joined together, I get some 'garbage' nodes I need to delete. I have tried to edit the layer and using nodes tool, selecting and deleting the nodes, but it seems impossible to do (nodes are not deleted at all!). Any idea on how to process this kind of clean-up?

EDIT : you can download the shapefile here for testing.

Answer

I tested your shapefile and used GRASS algorithm v.clean from the Processing Toolbox. I applied snap as the cleaning tool and used a threshold of 5 which seemed to have cleaned up your shapefile a fair bit.

This is your original shapefile loaded:

This is it cleaned using GRASS (notice the little yellow polygon that was dissolved originally but is now separate):

Just to show the difference in nodes, here was the original:

And here it is after being cleaned:

Hope this helps!