relative clauses - that which belongs to or is connected with her

that which belongs to or is connected with her (Merriam-Webster’s Learner’s)

'Though ‘which’ can be replaced by ‘that’ in relative constructions' they say, when it is placed after ‘that’ I wonder if aforesaid can be applied as well?

Answer

No, that which cannot be replaced with that that:

that which belongs to or is connected with her
that *that belongs to or is connected with her

See The Cambridge Grammar of the English Language p.1053:

(d) That which and all who: obligatory wh

[63] i. That [which we so carefully created] he has wantonly destroyed.
    ii. All [who heard her speak] were deeply impressed by her sincerity.

The very formal that which ("what") cannot be replaced by either that that or that, and when pronominal all applies to people who is required (cf. *all that heard her speak).

This conflicts with a rule sometimes given by prescriptive grammarians (although many which-hunters describe this case as an exception). Geoff Pullum has written about this on Language Log.