"8:14 is fulfilled in this dispensation, the sanctuary to which it refers must be the sanctuary of the new covenant. At the termination of the 2300 days, in 1844, there had been no sanctuary on earth for many centuries. Thus the prophecy, “Unto two thousand and three hundred days; then shall the sanctuary be cleansed,” unquestionably points to the sanctuary in heaven. GC 417.1"
"But Christ being come an high priest of good things to come, by a greater and more perfect tabernacle, not made with hands, that is to say, not of this building;"
The EGW passage interprets Daniel 8:14 as referring to the cleansing of the heavenly sanctuary. Hebrews 9:11 speaks of Christ as a high priest ministering in a greater and more perfect tabernacle (heavenly). The connection lies in the concept of a heavenly sanctuary and Christ's ministry within it.
"But the most important question remains to be answered: What is the cleansing of the sanctuary? That there was such a service in connection with the earthly sanctuary is stated in the Old Testament Scriptures. But can there be anything in heaven to be cleansed? In Hebrews 9 the cleansing of both the earthly and the heavenly sanctuary is plainly taught. “Almost all things are by the law purged with blood; and without shedding of blood is no remission. It was therefore necessary that the patterns of things in the heavens should be purified with these [the blood of animals]; but the heavenly things themselves with better sacrifices than these” ( Hebrews 9:22, 23 ), even the precious blood of Christ. GC 417.2"
"And almost all things are by the law purged with blood; and without shedding of blood is no remission."
EGW directly quotes Hebrews 9:22 to support the concept of cleansing the sanctuary with blood, both earthly and heavenly. This is a clear and direct reference.
"The cleansing, both in the typical and in the real service, must be accomplished with blood: in the former, with the blood of animals; in the latter, with the blood of Christ. Paul states, as the reason why this cleansing must be performed with blood, that without shedding of blood is no remission . Remission, or putting away of sin, is the work to be accomplished. But how could there be sin connected with the sanctuary, either in heaven or upon the earth? This may be learned by reference to the symbolic service; for the priests who officiated on earth, served “unto the example and shadow of heavenly things.” Hebrews 8:5 . GC 417.3"
"Who serve unto the example and shadow of heavenly things, as Moses was admonished of God when he was about to make the tabernacle: for, See, saith he, that thou make all things according to the pattern shewed to thee in the mount."
Ellen White directly references Hebrews 8:5 in this passage.